Combintorial inequalities

Table of contents :
A b s tra c t ....................................................................................................................... ii
A cknow ledgm ents ...................................................................................................... iii
V i t a ................................................................................................................................. iv
C H A PTER .............................................................................................................. PAGE
1 Introduction to Com binatorial In e q u a litie s ........................................... 1
1.1 The History of Combinatorial Inequalities ............................................... 1
1.2 Outline of the dissertation ........................................................................... 13
2 Quasi-Polynom ial S em i-L attice .................................................................... 14
2.1 Introduction to Quasi-Polynomial S em i-L attice .......................... 14
2.2 Statem ents and Proofs of Some Classic T h e o re m s ................... 22
2.3 Proof of the T h e o re m s ........................................................................... 23
3 The extrem e case of the Frankl-Ray-Chaudhuri-W ilson Theorem 30
3.1 A New Proof of the Ray-Chaudhuri-Wilson type Inequalities ............. 30
3.2 Characterization of the Extreme Case of the Frankl-Ray-Chaudhuri-
Wilson Inequality ............................................................................................. 35
4 M odular C om binatorial In eq u alities ........................................................... 53
4.1 Some Basic Modular Combinatorial Inequalities ..................................... 53
4.2 On the Mod-p Alon-Babai-Suzuki Inequality .......................................... 57
5 M iscellaneous R e s u lts ....................................................................................... 69
5.1 Special Cases of Snevily;s Conjecture ........................................................ 69
5.2 The Extreme Case of the Alon-Babai-Suzuki Type Inequality ............ 78
B ib lio g rap h y .................................................................................................................... 93

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COMBINATORIAL INEQUALITIES d is s e r t a t io n

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the G raduate School of the Ohio State University

By Jin Qian. B.S., M.S. *****

The Ohio State University 2000

D isserta tio n Com m itee:

Approved by

Professor D. K. Ray-Chaudhuri. Advisor Professor Thomas Dowling Professor Neil Robertson

tdvisor D epartm ent Of M athem atics

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UMI Number: 9994925

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© Copyright by Jin Qian

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ABSTRACT

In this dissertation, we study some com binatorical inequalities. First in C hapter 2 we generalize the concept of a polynomial sem ilattice introduced by Ray-Chaudhuri and Zhu [23] as a general framework in which to study the theory of combinatorial inequalities. We generalize it to a quasi polynom ial semilattice and then show by example that it strictly contains the polynomial sem ilattice as a special case. Also in C hapter 2 we show many of the classic combinatorial inequalities hold in the context of quasi polynomial sem ilattice. Secondly, the extreme case of Frankl-Ray-Chaudhuri-W ilson inequality is classi­ fied. It is shown in Theorem 3.2.1 of C hapter 3 th at the there is only one family th a t can attain the upper bound given in th a t inequality. Thirdly, a theorem is proved in C h ap ter 4 th at confirms a conjecture of Alon. Babai and Suzuki to a large extent. Finally, in C hapter 5 we give proofs to some special cases of Snevily's conjecture and prove a theorem which shows th at under a mild condition there is only one family that can attain the upper bound in the Alon-Babai-Suzuki inequality.

ii

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ACKNOWLEDGMENTS

First. I would like to tluink Professor Ray-Chaudhuri. who introduced me to this fascinating subject and spent a lot of time to teach me the basics of this subject and discussed the problem s with me. In the numerous meetings, he not only taught me mathematics but also English and many other things. I would also like to thank Professor Dowling and Professor Robertson for their excellent lectures on matroid theory and graph theory. I would like to thank my fellow graduate student Yared Nigussie for many helpful discussions. Last but not the least I would like to thank my wife leilei for her cheerful presence when this dissertation is being w ritten.

iii

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VITA Oct. 20. 1970 .............................................

Born in laian. P. R. China.

1992 .............................................................

B.S.. Nankai University. T ianjin. P. R. China.

1995 .............................................................

M.S.. Ohio S tate University. Colum bus.

1994-present ...............................................

G raduate Teaching Associate. T he Ohio State University

PU B L IC A T IO N S

On Frankl-Fiiredi Type Inequality Combinatorial Inequalities for Quasi-Polynomial Sem i-lattice Extremal Case of Frankl-Rav-Chaudhuri-W ilson Inequality On the Mod-p Alon-Babai-Suzuki Inequality

iv

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FIELD S OF ST U D Y

Major field: M athem atics Specialization: Combinatorics Studies in

:

Combinatorial Inequalities

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TABLE OF CONTENTS

A b s t r a c t .......................................................................................................................

ii

A c k n o w l e d g m e n ts ......................................................................................................

iii

V i t a .................................................................................................................................

iv

C H A P T E R .............................................................................................................. 1

PAGE

In tr o d u c tio n to C o m b in a to ria l I n e q u a l i t i e s ........................................... 1.1 The History of Combinatorial In e q u a litie s ............................................... 1.2 Outline of the d is s e r ta tio n ...........................................................................

13

Q u a si-P o ly n o m ia l S e m i - L a t t i c e .................................................................... 2.1 I n tr o d u c tio n t o Q u a si-P o ly n o m ia l S e m i- L a tt ic e .......................... 2.2 S ta te m e n ts a n d P ro o fs o f S o m e C la ssic T h e o r e m s ................... 2.3 P r o o f o f t h e T h e o r e m s ...........................................................................

14 14 22 23

T h e e x tre m e c ase o f th e F ra n k l-R a y -C h a u d h u ri-W ils o n T h e o r e m 3.1 A New Proof of the Ray-Chaudhuri-W ilson type In e q u a litie s ............. 3.2 Characterization of the Extreme Case of the Frankl-Ray-ChaudhuriWilson I n e q u a lity .............................................................................................

30 30

4

M o d u la r C o m b in a to r ia l I n e q u a l i t i e s ........................................................... 4.1 Some Basic M odular Combinatorial In e q u a litie s..................................... 4.2 On the Mod-p Alon-Babai-Suzuki I n e q u a lity ..........................................

53 53 57

5

M isc e lla n eo u s R e s u l t s ....................................................................................... 5.1 Special Cases of Snevily;s C o n je c tu r e ........................................................ 5.2 The Extreme Case of the Alon-Babai-Suzuki Type In e q u a lity ............

69 69 78

2

3

B i b l i o g r a p h y ....................................................................................................................

vi

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1 1

35

93

CHAPTER 1 INTRODUCTION TO COMBINATORIAL INEQUALITIES

In the first section we briefly talk about the history of "Theory of Com binatorial Inequalities" (or "Combinatorial Inequalities” for short) along with some new re­ sults obtained in the dissertation. In the second section we sketch an outline of the dissertation.

1.1

The History of Combinatorial Inequalities

In our presentation, all "old” theorems are attrib u ted to the original discoverers. Throughout this dissertation, unless otherwise stated , n is a positive integer. k . s . r . t are positive integers th a t are less than n. i ' is a set of n elem ents and /„ = { 1 . 2 . • • • .a} for any positive integer a. The theory of combinatorial inequalities is a very active branch of Combinatorics. It begins with Sperners interesting theorem [25] published in 1928. T h e o r e m 1.1.1. (Spemer) Let F C F(Y). I f fo r any distinct E . F E F . E g F , then |JF|
t fo r all E . F

6

IF with E ^ F. then \F\
n0(t. k). They also estim ated that the n0{t.k ) < t + (k — t ) ^ ) 3. In 1976. FrankI proved that nQ(t.k) = [t -r 1 )(A: - t. -f I) for t > 15 and in general th a t no(k.t) < rk(k — t) for some constant c. In 1984. W ilson pnwed that n0(k .t) = (t + 1 )(k — t +

1

) in the remaining cases

t = 1.2.3. • • • .14. which completes the search for n0(t. k). More precisely, he proved the following theorem . T h e o re m 1 .1 .5 . ( Wilson) Let n > (t 4- \ )(k — t + 1). I f J- is a family of k-subsets of Y with the property that \E fl F\ > t fo r all E . F If n > (f + I )(A: — t 4- 1 ) and \(F\ =

6

E with E

F . then \F\ < (““ *).

. then J- consists o f all the k-subsets which

contain a fixed t-subset. In 1975 R ay-Chaudhuri and Wilson [22] generalized Theorem 1.1.3 to m ultiple intersection sizes. Before stating their theorem , let us introduce some notations which are im portant throughout the dissertation.

Let £ be a finite set of non-negative

3

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integers and T be a family of subsets of Y . We call T an L-intersection family if \E H F\ € L for any E . F G T with E ^ F. We also call L the intersection size set. Let u. k. A. t be positive integers. For any set IF with v elements we call a family 7Z of A-subsets of IF a t — (v. k. A) design (or t design for short) if for any subset .4 of IF with size t. the num ber of subsets (called blocks) in 7Z that contain A is A. A 3( 3. and let |.4A B | = d for any two distinct blocks A and B.

If

|£J| = r = 2d, then B is equivalent to the family {/„} U A . where A is a family of c — 1 blocks o f a Hadamard 3-(r. v/2. v/A — 1) design with no two blocks being complementary. I f \B\ = v ^ 2d, then B is equivalent to the block-set of a symmetric 2 - { r . k . \ ) design. I’nlike in ease of the Rav-Chaudhuri-W ilson inequality, where the extreme eases were elassified. the Frankl-Ray-Chaudhuri-W ilson inequality does not come with a elassifieation of the extreme cases. In C hapter 3 of this dissertation, we will show that the only family that can atta in the upper bound under the condition in the Frankl-Ray-Chaudhuri-W ilson inequality is the obvious family, namely the family that consists of all the subsets of Y with sizes < s. O ur result will be stated and proved in a more general context. The following is a special case of the result. T h e o re m 1 .1 .1 0 . Let T C P ( l') be a family of subsets o f Y . L C N u {0} be a finite set with s elements. If IF is an L-intersection family, then \!F\ = 5Zj’_u (”) if and only if L = { 0 . 1 . 2. • • • . s - 1} and T = U Lo Theorem 1.1.7 is very robust in th a t by varying the condition in the theorem, one can get many improvements. Here we list two such examples. By using additional information on |{ |E | : E e F } | , Alon. Babai and Suzuki [1] improved the above theorem and obtained the following theorem. T h e o re m 1 .1 .1 1 . (Alon, Babai, Suzuki) Let J- C F{Y) be a family of subsets of Y . £ C N u {0 } be a finite set with s elements, K be a set o f nonnegative integers with

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| A'| = r and k > s — r + VEer.tium m

1

. Vk £ K . If T is an L-intersection family and |£ j £ A'.

< c ) + ( ; j

+ --- + U J -

Recently R am anan [21] gave a very elegant proof of an im portant special case of the Frankl-Fiirprli conjecture. This special case of the conjecture can he thought of as an improvement of the Frankl-Ray-Chaudhuri-W ilson Inequality when the inter­ section size family L = {1.2. • • • . s}.

Theorem 1.1.12. (Ramanan) Let L = {1.2,-* - . s}. If IF C P (l’) is an L-intersection family, then \IF\ < X7=o (” 7l) In the proof of this theorem. R am anan used quadratic forms and a m ethod called "system of linear forms” . This m ethod is easier to use compared to the incidence' m a­ trix method even though fundamentally they are equivalent and it forms the frame­ work in which m any of the proofs of our new results in this dissertation are carried out. In a different direction. Frankl and Wilson [1 1 ] also proved the following m odular version of the above Ray-Chaudhuri-W ilson Inequality. Before stating it. let us intro­ duce some notations here. If p is a prime number. L is a set and a is a non-negative integer, we say a £ L (mod p) if and only if there exists I £ L such th at a = I. (mod /;). Similarly if a £ L (mod p) if there does not exist such an I

6

L.

Theorem 1.1.13. (Frankl. Wilson) Let p be a prime number, s be a positive integer with s < p and L be an s-subset of {0,1, • • • ,p — 1},. IF C !Pfc(V) be a fam ily of k-subsets of Y fo r some k £ N with k > s and k £ L (m o d p). If for any E. F £ IF with E 7 ^ F . \E n Aj £ L(mod p), then \F\ < ("). 8

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It's very clear th a t this theorem reduces to the Ray-Chaudhuri-W ilson inequality when p > ri. To see th a t it is also more powerful than Theorem

1

. 1 .6 . let us take

a look at a simple example. L\ = {1.6}. k — 7. then the upper bound given by Theorem

1

. 1 . 6 on \F\ is (") but the upper bound given by Theorem 1.1.13 is (")

since 1.6 G L (mod 5). 7 £ L{mod 5) for L = {1}. As a corollary to the above theorem. Frankl and W ilson obtained the following theorem [1 1 ]. T h e o re m

1.1.14. (Frankl. Wilson) Let n = 4p — 1

J" Q) is an L'-intersection fam ily , 1

fo r some prime number p. If

where L' = {0. 1.2. ••• . p — 'l.p.p +

. • • • . 2 p —2 }. then \F\ < ( 'J " 1). This theorem follows easily from the Theorem 1.1.13 since if we take L = {0. 1. • • • . p —

2}. then 'F is a m od p L-intersection family. This theorem was used by Frankl and Wilson [1 1 ] and K ahn and Kalai [16] to give two beautiful results in Geometry. The unit distance graph on Rn is the graph whose vertex set is Rn and any two vertices are adjacent if and only if the distance between them is 1. The chromatic number of a graph is the minimal num ber of colors

th a t one would need to color

vertices of a graph so that adjacent vertices receive

different colors. Larm an and

Rogers [14] conjectured that the the chrom atic number of the unit distance graph on Rn is exponential in n. Frankl and W ilson proved this conjecture by applying the above theorem. For n of the form 4p — 1 for some prime num ber p. they considered the subgraph of the u n it distance graph on Rn whose vertex set consists of all the 0-1 vectors in Rn of weight 2p — 1 . Here a 0-1 vector in Rn means a vectors whose

9

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components are either of the vector.

0

or

1

and the weight of a vector is the sum of all com ponents

By Theorem 1.1.14. for any coloring scheme, each color class (the

collection of vertices the the same color) is a t m ost ("jpj1) and so overall one needs at Mp-1 \ ('lp_l) least colors. It is not hard to show th a t is exponential in p and hence in v tJ —» /

. P

. .

n. Next one uses a density theorem for prim e numbers to generalize it to general n. In 1993 J. Kahn and G. Kalai [16] employed the above theorem to give a disproof to a long-standing conjecture by Borsuk [4]. Borsuk's conjecture states th a t any set in JR'1 of diam eter d can be partitioned into n + I subsets each with diam eter less than d. Here the diam eter for a set means th a t maximal distance among two points in the set. Theorem 1.1.6 is generalized by Alon. Babai and Suzuki in [1]. We state their theorem as T h e o re m 1.1.15. (Alon. Babai. Suzuki) Let L. K be two subsets of {0. !.••■ . n} with \L\ = s. |A"| = r. r < s and kt > s — r + I for all kt L-intersection family with \E\ € K . VE € T , then \IF\ < (”) -I-

6

A\

If T is an

-t- • • • +

t).

This theorem is more powerful than Frankl-Ray-Chaudhuri-W ilson inequality in some situations. For a concrete example, let Y = { 1 , 2. • • • , 9}. T C IP,,)!') |J Pr>0*) be an {0 , 2 ,3 [-intersection family. By the Frankl-Ray-Chaudhuri-W ilson inequality. I-^I < (3 ) + (2 ) + Ci) + (o)- bu t by the Alon. Babai and Suzuki's inequality. \E\ < (3 ) + © • It is not hard to see th a t if T — Ui=a-r+ i

T satisfies the conditions in

the above theorem and it achieves the upper bound. In C hapter 5 of the dissertation

1 0

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we will show th a t under the mild condition th a t s
€ X . {z £ X t\z < F\. z < F>} = {z e X,\z < Fi A F,}. From a) of the definition, we conclude th a t { r G A',|c < F[. z < F2}\ = {r G

X t\z < F l A F > } \ = M I F i A F2|). Recall that \F\ is m, if F G A',.

The following examples taken from [23] are quasi-polynomial seini-lattices which are also important com binatorial objects. Examples: 1) Johnson Scheme. Let V be an n-element set and A'* be the set of all i-element subsets of V' 0 < i < n. Then X = U"_0 A'’,-, with inclusion as the p artial order, is a semi-lattice.

Let mi = i, f t(x) = (■'), a polynomial of degree i in the

variable x. It is easy to see th a t (AT. < ) is a quasi-polynomial sem i-lattice. 2) q-analogue of Johnson Scheme. Let V be an n-dim ensional vector space over a finite field GF(q), A; be the set of all i-dimensional subspaces of V.

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0

< / < ri.

Let m t = ql, f i ( x ) = [f]9, a polynomial of degree i in the variable x.

Then

X = U-L0 A't is a quasi-polvnomial sem i-lattice with inclusion as the partial order. •J f

l XUt Ul lOOt oij

LLt

1

tU l

O “ C t 0 i A l ( J i l L .I L L ,

= {(T./i) | L C {1,2

We define A', },

►► u L

< i < ri.

A'o = {0 }, where

n}. 0

W u O iO

O 1.3

|L| = i.

c t J./U 3 1 LI Vt_* l l l L U t i j C i .

h : L —» If'

a map

is taken to be the least element, and

X = U-L0 A'i. (L \ . h \ ) < (L2, h 2) if and only if L\ C L2. and h2\Ll = h-i- Then (A'. M\ a linear

Let A = U£_0 A,. V(L’i, hi). {i'2. h->)

(L’i.hi) < (U2, h 2) if and only if L\ C U2 and h2\ui = hi-

6

A', define

Then (A*. < ) is a

quasi-polynomial semi-lattice, with m, = q ' . f f x ) = [f]7. 5) O rd e re d Design. with n < s. We injection}.

1

A' = UjLoAj.

Let 11' be an s-element set and V be

an n-element set

define A'; = {(L . h ) | L C V,\L\ = i.

h : L —> U' an

< i < n. A'o = {0}, where

0

is taken as the least element, and

V (Ll? hi), (L2, h2) € A', define {Li, hi) < (L 2. h 2) if and only if

L\ C L2 and h2\ i x = /q. Then (A'. < ) is a quasi-polynomial semi-lattice, with rrq = i. fi{x) = (*).

6

) q-analogue o f Ordered Design.

Let W be an s-dimensional vector space and

18

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r

be an n-dimensional vector space over a finite field GF{q) with n < s.

Define A', = {(U.h) \ U C V . d i m ( U ) = i. h : U —> IF. a non-singular linear transform ation }. 0 < i < n. Let X = U"_0A'j.

V(L’l.h i). (U2. h>) G A . define

{U[.hi) < (U-y.ho) if and only if L\ C U2 and /i2 1c/t = hi-

Then (A”. , x 2 G A \, i/•>€

6

A'3. |{r G AT|0 < :
< y-2. Suppose affine lattice is a polynomial sem i-lattice in the sense of [23](cf §1). Then \{z G A'^0 < c < f/i}| = |{c G A 3 |x 2 < c < y2}|.

(2.1)

It is not hard to see th at the left hand side of (2 . 1 ) is q. To find out the right side, we observe that if x 2 = v + If' for some vector subspace W of V and c G I ', then |{r G A'3 |x 2 < c < y2}\ = |{c - u G A'3 |x 2 - v < z - v < y2 - c}|.

(2.2)

where S — v := {s — u|.s G 5} for S C V. If we take x 1., = x-> - v G X 2, z' = c — u G A' 3 and ij'2 = ;/■> — r G A'.j. we see that x'.2, z '. y'2 are vector subspaces of dimension 1. 2. 3 respectively because an affine subspace containing 0 must also be a vector subspace. The right hand side of (2.2) is easily seen to be 20

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< z '< i£ } |

= the number of 2 -dimensional vector subspaces contained in

ij!2

and

containing x 2. = the number of 1 -dimensional vector subspaces of the the 2 -dimensional quotient space ij2' / x '.2 !)/(< / -

= (q2 -

i)

= 7+1. So the right hand side of (2.1) is equal to q + 1 . This implies that affine lattice does not satisfy (2 . 1 ) above and therefore it is not a polynomial semi-lattice. Fact 2: Affine lattice is a quasi-polynomial semi-lattice. Proof By the argum ent used in the proof of fact 2. we see th a t f tJ = f tJ =

0

if i > j : and for i < j . i >

fa

1

1

if i = 0:

.

= _

qJ ~ l { qJ ~ l -

- < ? ) • • • [ q J~ l -

then |F] < (").

Our theorem 2.2.4 proves a special case of Snevily’s conjecture for quasi-polvnomial semi-lattice. However this conjecture of Snevily is still open.

2.3

Proof of the Theorems

Throughout the rest of the chapter, we have the following conventions: 23

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1

. The em pty product is defined to be 1, i.e. n*er, a. • • • . Fm} be such th a t /( F t ) < /(F>) < ••• < l(Fm). Here we recall th at /(Fj) is the height of the element F,. We need to show th a t m
s — t + 1. VA, 6 K). So the m atrix (fi,tkl) becomes a m atrix whose (u. i)-entrv is ((x) -I

27

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b csgs{x)

has t, distinct roots m w h e r e i = 1 ,2 . • • • ,t. Since the polynomial is of degree < t — L. it is equal to the zero polynomial. Therefore c3 _t+ 1 = cs _ t + 2 = • • • = r, = 0. So the matrix is not singular. This finishes the proof of the claim and hence th at of lem m a 2.3.1 and theorem 2.2.3.

□

P r o o f o f th e o r e m 2.2.4. We consider the \Jr\ by |.Y.,| incidence m atrix M whose rows are indexed by elements of T and whose columns are indexed by elem ents of A',. For .4. € T . S G A’,, the (.4. S )-en try of M is defined to be I if either .4 G A\ and 5 = .4 or /(.4) > s. S G A’s and 5 < .4. It is defined to be 0 otherwise. Observations: It is clear from the above definition of A/ th a t (L) if the (.4. S)-entry of M is 1 then 5 < .4. (2 ) each row has at least one non-zero entry and (3) for .4 G F w ith .4 6 A',,, u > s. the row corresponding to .4 has /,( t» ,J > L non-zero entries( see the definition of quasi-polynomial sem i-lattice in §1). Claim: For .4. B G T with .4 ^ B . the (.4, B)-entry in M M 1 is 0. Proof of the claim:

Suppose (.4, i?)-entry of M M T is > 1.

Then there exists an

5 G .Y, such th a t both (.4. S)-entry and (5 .5 )-e n try of M are 1. (1). 5 < . 4 a £ . so S G U!=o

By observation

since T is a {0,1, — s — l}-intersection family. This

contradicts the fact that S G A's.

□

From the above claim, it is clear th a t M M 7 is a diagonal m atrix, and it is also clear that the diagonal entries are non-zero by observation (2) above. So M M r is a non-singular \F\ by |JF| matrix and \T\ = rank(A/A/r ) < rank(A /) < |A’,|. Now suppose \F\ =

| Al s |, s o

A/ is a square matrix. It is clear th a t each colum n of

A/ can contain a t most one non-zero entry, otherwise M M 7 would not be a diagonal 28

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m atrix.

So the to tal number of l's in S I is < |A'S|.

By observation (2) each row

contains at least one non-zero entry and so the total number of l ’s SI is > \T\. Since \IF\ = |A*,|. the total number of l's in S I should be exactly |JF| and hence each row of SI contains exactly one non-zero entry. This means th at A G A\ for any A G T .

□

So T C A',. But \F\ = |A's|. so F = X a.

29

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CHAPTER 3 THE EXTREME CASE OF THE FRANKL-RAY-CHAUDHURI-WILSON THEOREM

In this chapter we give a new proof of the famous Frankl-Ray-Chaudhuri-W 'ilson Inequality and then characterize the extreme case of it. In fact we show that the upper bound in the Frankl-Rav-Chaudhuri-W ilson Inequality can only be achieved when the intersection family is the trivial family consisting of all the subsets of X of size less than or equal to s and the intersection size set happens to be { 0 .1. • • • . s - I }.

3.1

A New Proof of the Ray-Chaudhuri-Wilson type Inequal­ ities

In this section we first give a new proof of Theorem 2.2.1 which is a generalization of the Frankl-Ray-Chaudhuri-W ilson Inequality in the context of a quasi-polvnomial semi-lattice. This new proof is inspired by R am anan’s “System of linear equations" method [21]. Secondly we present the new proof of a theorem th a t generalizes the Ray-Chaudhuri-W ilson Inequality to the context of quasi-polvnom ial semi-lattice. This proof is based on the one given by R am anan [21].

30

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Before we give the proof, let us introduce some notations which will be used in the rest of this dissertation. Let (A”. .••• . L} be an selement subset of /„ U {0} with l\ < l-> < ■■■< ls. E C A' is called an L-intersection family if for any E ^ F (z T we have l(E A F) G L. where /(•) is the height function associated with (-Y, < ). For each F G T . we associate a variable X f. For I € X , we define the linear form L, by Li =

^ x f = ^ 2 '\r{F)x r F€F.F>1 F€?

where for any / . F 6 A’. A /(F) is defined to be 1 if F > I and 0 otherwise. In the following we give a new proof of theorem 2.2.1. Recall th at the theorem 2.2.1 states: Let (A. < ) be a quasi-polynomial semi-lattice of height n, L C [n U {()} with \L\ = s. If T C .Y is an L-intersection family, then \E\ < \A’o| + |A'i| + •••-(- |A'S|. P ro o f. Consider the system of linear equations: | l / =0.

where I runs through A'o U A'i U • • • U A',.

(3.1)

Let (v p ) . F G T (henceforth abbreviated as {up)) be any solution to the above system of linear equations. We hope to show (vp) is the trivial solution, which would imply th a t |A'o I + |A'i| + ■■■+ |A'a| > \T\ because by linear algebra, if a system of linear equations has the trivial solution only, then the number of equations is > the number of variables. So it is enough to prove the following lemma.

31

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L e m m a 3.1.1.

We keep the same assumption as in theorem 2.2.1. If F

is an L-

intersection family, then the following system of linear equations can have the

trivial

solution only. \ l, =

.

I € ( J A'j.

0

1

(3.2)

1=0

P ro o f. Suppose ( v e ) is a solution to the system of linear equations. It is enough to show that (cE) -= 0. where 0 is the all 0 vector in R ^ 1. Suppose on the contrary. 7^ 0- Define ./ = {1{E)\E € IF. l'e

0} and let j 0 be the largest number in ./.

So there exists E0 € F . such that /(Eo) = jo and ve0 ^ 0. Let

/M =

II (x ~ m O l, / , ( | £

a

£„|).

i= 0 32

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hence f ( \ E A £ 0|) which is equal to th at of the right hand side. Specializing x E = t'c, V £ d 5 > -

6

we have (3.3)

J 2 M M )A /(£ o ) = ^ / ( |£ a E o |) l - e .

Let us consider the right hand side of the above equation. For any £ € F with E ^ E0, if /(£ ) < j 0. then / ( E a £ 0) < I(Eo) = jo and so / ( |£ A E 0|) = 0: if /(£ ) > jothen by the definition of J and j 0, vE =

0

. Overall, the right hand side of the above

is equal to /(|E o A Eo|)c'e 0 = f { m jo)uEo. The left hand side of (3.3). however is equal to zero because (i:E) satisfies L/ = for all /

6

0

A’n U A j U ■• • U A’,, and d < s by the definition of d. So 0 = f(rrijn)rEl) and

we get rEn =

0

. since f { m j 0) ^ 0 by the definition of / . This contradiction completes

the proof of this lemma and hence theorem

2

.2 . 1 .

□

Next wo present a new proof of theorem 2.2.2 which is a generalization of the Frankl-Ray-Chaudhuri-W ilson Inequality in the context of quasi-polynomial semi­ lattice based on the one given by Ram anan [21]. Recall th a t theorem

2

.2 . 2 states:

Let (X. . we can apply the induction hypothesis

(3.6) to the first term of

(3.6) (denoted by I) and wo have I = ( - l ) ( x - rrij) E * = o l- l )‘uiU' -

1

) • • -(x - m;+1+t_[)

= Ei=o(“ l ),+lM * - n i j ) { x - ni j +i ) ■■• ( x - mJ+1+1_ l ) for some positive rational numbers u„. u ,_ i. • • • . u0. Then we use the induction hypothesis on the second term (denoted by II) and we have II = (mtl - rrij) E « = o ( ~ ^ ‘^(x - rn} ) ••■ (x - mJ+1_[)

= E L o { - l Y ( m h “ m j ) v i { x - rrij) • • • (x for some positive rational numbers vs, os_i, • • • . v0. Adding up I and I I gives

( - l ) ( x - mJ)[(-l)-'(x - m h ) • • • (x - m*1+1)] + (m([ - m j ) [ ( - l ) s(x - m,,) • • • (x = E S l - 1)1^ ^

~ mj) •••(-r “ mj+ - 1 )

where d.s+i = us.

d, = us_i + (tn^ — m. j ) vs , 39

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= its—2 + {m tl - m j)-u s_ lt

' / o = ( w ; , - rrij)vQ. So

• • • . r/ 0 are positive, proving the claim and therefore the lemma.

□

Further by using the sim ilar m ethod, we have the following corollary which is very useful in the rest of the proof of Theorem 3.2.1. C o ro lla ry .

3.2.1 Let (.V. < ) be a quasi polynomial semilattice, s a positive inte­

ger with s < n and li .l 2.-- - . l s be s non-negative integers in {0 . L.2 . - 1- . n } with {/L, /_). • • • ,/,} ^ {0. 1. • • • . -v — 1}. There exist s -F 1 non-negative rational numbers Iiq. h[.■■ ■. bs with bs-i >

0

such that

S

^ ( - l y b i f d - v ) = ( - l) * ( x - m h ){x - m h ) . . . ( x - m ,J . i= 0

P ro o f. W .L.O.G. we suppose 11 < /■ > ,< •••


0

. then the corollary follows

from the above lemma. So in the following we assume Zt = 0. We let a be the largest integer such that {/t . /•).•• • . ls}

= 0. l2 = 1. • • • . ln = it — 1 . Since

{0, 1. • • • . s — 1 }, we have u < s. Using the claim in the above proof

of Lemma 3.2.2 with j = u and Zu + l as /[, there exist positive integers dn. d„+l. • • • . ds with ^ ( - i ) ‘cZu+l(x - m u)(x - m u+l) ■■■{x - m u+i_!) 1=0

= ( - l ) A_u(:r - m lu+l)(x - m K+,) ■■■{x- m ,J .

40

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Now we m ultiply both sides by ( —1)u(a: —m0)(x —m i) • • • (x - mu_ i ) (which is equal — mi.,) ■• ■(x — rntu) by the definition of u) and we have

to (—1 )u(x — S — tt

^

(

-

1

Y +udn+t{ £ - m Q) { x - r n l ) ■■■

t) =

)s( x - m fl) ( x -m i,) • • • ( .r - m ,J .

( - 1

i= 0 i.e. X > D H ( x - rn0)(x - mi) ■• ■(x - " h - 1 ) = ( —l ) s(-r - mf,)(x - m,_.) ■■■(x - m ,t ). t= 0 with do = d\ = ■■■ = r/ , , - 1 = 0. By the definition of quasi-polynomial semi-lattice, this implies th at .S

] P ( - l ) ,(/'/ 1 ( .r )(-l) ''i(.r - m ,,)(x - m h ) ■■■{x - m ,J . i= 0

here dt = d'c,. i =

0

. 1 . • • • . n. wrhere c\'s are the positive constants associated with

the quasi-polynomial semi-lattice. This completes the proof of the corollary.

□

L e m m a 3 .2 .3 We keep the same notation as in theorem 3.2.1 and lemma 3.2.2. If we let g(x) = (x — m j,)(x —mi.,) ■■■(x —m jJ. then we have: .-i

£ ( - l ) ' & , £ Lj = £ ( - l ) ‘S( | £ | ) 4 1=0 /e.v. Eer

(3.7)

L e m m a 3 .2 .3 . We keep the same notation as in Theorem 3.2.1 and Lemma 3.2.2. If we let g(x) = (x — m(t )(x — m /,) • • • (x — m i, ), then we have: S

£ ( - 1 ) ' 6 , £ Lj = £ ( - l ) * 1=0 /e.v, ecjf

9

( |£ |) : r |

41

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(3.8)

P r o o f. We regard both sides as quadratic forms in x e s, E € T and try to show that the corresponding coefficients are equal. For example, for E, F € T w ith E # F. the term L j contributes a term 2x/ri’/r if and only if I < E A F. Therefore the coefficient of x e X e in the right hand side of (3.8) is 2

C—

Ai^l) (see the remark 3 in C hapter 2 §1 ) which is equal to

2(—1)s(|£7 A F | - m tl){\E A F | - m ,.,). . . ( |£ A F | - m*J by lemma 3.2.2. Since T is an L-intersection family with L = { L . L {nn,. m u

(3.9) /,}. |F a F | €

m /,} and so the product in (3.9) is 0 . Obviously the coefficient of x e X f So the coefficient of x Ex F in the left hand side is

in the right hand side is also 0.

equal to th at in the right hand side. Sim ilarly the coefficient of xjr in the left hand side is 52*_0( —1)16 , / , ( |£71) for the same reason as above. By lemma 3.2.2 it is equal to ( —1)S ( |£ l ) - i

EeF

/€.V ,

We divide both sides by ( —I ) 4 and move the left hand side to the right hand side. So we have

°=

£ i t E { 0 . 1 . . . s —t is od d

£ )£ /((■ * )) + £ > ( | E | ) 4 l€X,

(s in

EeX

If £ is a gap element, then by the condition in this lemma, we know vE =

.

0

If E is not a gap element, then it is clear that g(\E\) > 0 where we recall that g(x) = (:r — m (,)(x — m ^) •■•(x — mja). So in any case g(\E\)uE > 0. This means that the right hand side is a sum of nonnegative terms. Obviously if /(£") > /,. i.e. \E\ > mit , then 0. and so vE = 0.

Recall th a t /(.) is the height function

of the quasi polynomial sem i-lattice (X , < ) defined in C hapter 2 §1. The equation (3.11) alsoimplies th at L[((uE)) = 0 for I GA'j, i = s — 1, s —3. • • •. So L [ ( ( v E )) =

43

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0

for all I G X 0 U A'i U • • • U X s. In particular. LQ((vE)) = 0 where 0 in the subscript is the least element of A'. To show th a t vE = 0 for all E G E . we assume the contrary. Define J = { l ( E ) \ E G E. rE ^ 0}. Let j Q be the largest number in J. Suppose E G E is such that 1(E) = jo and vE ^ 0. We fix such an element E. By the results in the previous paragraph, we have jo < /,. In the following, we distinguish

2

cases:

Case 1 . Suppose j 0 < L . Since E is an {ly. /•>. • • • . /.^-intersection family, we have I ( E a F) > / t > j 0 = 1(E). VF G E . and so F > E or F = E. If F # E. then F > E and /(F ) > 1(E). Therefore vE = 0 by the definition o f ./ and j 0. Further because 0 = L0 ((c/r)) =

'F = 'F -

we have uE - 0 . a contradiction. Case 2. Suppose /t < j 0 < /,. Let u = | {/|/t < 1(E) }|. Since /o. / i . • • • . f u form a base of the vector space of poly­ nomials of degree < u. there exist real numbers c0, Ci, • • • . cu such that

c,/,(.r) =

h(.i:). where h(x) = Yli,^i{E)(x ~ m t E and F ^ E. then 1(F) > 1(E) = j Qand by the definition o f./ and j 0. vF =

0

Since h(\E\) #

. So the right hand side of (3.12) is equal to h ( \ E A E\ ) v e = Ii(\E\)l'e. 0

. we get ve =

0

. a contradiction. This proves this lemma .

□

As a consequence of lemma 3.2.4. if L consists of consecutive positive integers, then

\J-\

< |A , | + |

_ •>| + • • • + | A i —2 *-/gj I < | A | + |A.,_i| + • • • 4- j A ot • ,s

In the next few lemmas, we show th at if the numbers in L are not consecutive, then |F | < |A'()| + |A't | + • • • + |A'S|. We do it in two steps. First we deal with the case where L is not consecutive and L ^ {0,1. • • • .s — 2,ls } for any /, > .s in Lemma 3.2.5 and then in Lemma 3.2.7 we deal with the case where L = {0 . 1 . • • • . .s - 2. /,} for some ls > s. Before stating Lemma 3.2.5, let us make an observation. If E is a gap element, then for any F 6 F F / £ . l( E A F ) < 1(E) and so l(E A F ) < 1(E) since 1(E)

L.

L e m m a 3.2.5. Let (AT. s. then \ T\ < |A'0| + lA'^ 4

b |ATS_->| + |A'.,_i| 4- |A',|.

45

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P r o o f o f L e m m a 3 .2 .5 . We consider the following system of linear equations jz .r = 0

V / G A'o U A't - • • U A,_> U A',.

(3.13)

If the system (3.13) has the trivial solution only, then \F\ = th e number of variables < th e number of equations = I-V)| + | A 11 -+*••• +

4- |A.,|

< IA o | 4- IA '! j 4- • • • 4- |A.,_->| 4- |A .,_ i| 4- | A., |

Which would prove lemma 3.2.5. So it is enough to prove the following lemma which also has many applications in the last chapter. L em m a

3.2.6. We keep the same assumptions as in lemma .1.2.5.

The system of

linear equations (2.13) has the trivial solution only.

Proof. Suppose (l'e)ei=.f is a solution. We hope to show (up) is the all-zero vector in !R|jrL By Lemma 3.2.4. it is sufficient to prove th a t v£ = E e E . i.e. v[r =

0

0

for every gap element

for any E € T with n ^ i d ^ l “ m(,) < 0- We fix such a gap

element E € E . it is clear that for such an E. there exists i € { 1 . 2. • • • . s — 1 } with lt < 1(E) < li+i. Note th a t since /, > i — 1, we have 1(E) > i. We distinguish two cases. Case 1 . 1(E) ^ i 4- 1. Since i < s — 1. we have two subcases: Case 1 . 1 . i = s — 1 . In this situation, we have 1(E) > i 4-1 = s. So {/t . l2. • • • . /,_ t. 1(E)} ^ {0 . 1 . • • • . s — 1 }, by Lemma 3.2.1 we know 46

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^ f o ( r nt l )

fi(rnh )

•••

/ S_ 2( m / J

fs(mh ) ^

.4 :=

is non-singular.

Minin)

/i

Mrrn,^)

) •••

f d \ E \ ) •••

\ M\E\)

(

f - ^ ( \ E\ )

f J\ E\ )

\

«o rii

So there exists a =

such that

7

V 0

.4a =

(3-14)

\ lj As before, for any I

6

X . E € T . Xr{E) is defined to be

1

if I < E and

0

otherwise. We next prove X j=o

In fact, for any F

6

> £

L [ \ i ( E ) -I- a s _ L

re\j

L [ \ [ { E ) = xe-

(3.15)

iex.

f . the coefficient of x p in the left hand side of the above equation

is 5 —2

] T a j f j d F A E\) + as^ M \ E A F |) j=o

47

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( 3. 16)

By the observation made before Lemma 3.2.5, if F ^ E. then l(F A E) < l[E) and l(F A E) € {H-l-i- • • • . C-i}- and so (3.14) implies th a t (3.16)= 0: if F = E. then s-2

^ a J/ J ( | F A £ | ) + f l , _ l/ s ( | F A £ | )

j= o .1 - 2

= 5 > , / , ( | £ l ) + «,_,/,(|E|) j

= ()

=1 This proves

by (3.14).

(3.16) and hence (3.15).

If we specialize x p — up in (3.15)forall F € F . the left hand side will be which we get vp =

0

0

. from

.

Case L.2. i. < s — 2. Let f ( x ) = (x —m/, )(x —mp) ■■■(x — mi J . then there exist an. =

1

. •••. /„_! -- a — 2 . and so

••• . £} = {0 . 1 .-- - .s -

2

./,.}. which is a

contradiction to our hypothesis. So i < s —2 and therefore E G .V0 UA'iU- • • A'.,_-2 uA\.. This implies L/r is one of the equations in (3.13) and so L e (( l'f )) = get that

0

. Further we

= x e by the same argument as in the proof (3.15). So we have vE = 0.

This completes the proof of lemma 3.2.6 and hence th a t of lemma 3.2.5.

□

L em m a 3.2.7. Let (X. < ) be a quasi-polynomial semi-lattice and J- C .V he an Lintersection family. If L = {0. 1, • • • . s —2. /.,} where

> s. then |JF| < | A"o| + |Xy \ -f-

• • • -t- |A',j. P ro o f. Again we have 2 cases. Case I. A',_[ C T . Suppose that T contains an element E with 1(E) > s. Let A be any (dement in A',_i with A < E. then l(A A E) = s -

1

. This contradicts the fact th a t E is an

L-intersection family w ith s — l g L . This shows th a t E does not contain an element E with 1(E) > s. So |^ | < |A’o| + |A'l| +

b

which proves Lemma 3.2.7.

Case 2. A's_! g E. In this case. | E D A's_[| < |AA-i|. We want to show that the system of linear equations /

L, = 0

V/ G A'q U X i ■• • U A' 3 _ 2 U A', (3.18)

I £/ = 0

V/ G E . l ( I ) = s - 1

49

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has the trivial solution only, which will show that

\ F \ < | A'o | + \ X i \ ■■■ +

| A S_ , | + | A ', | + \ T n A ', _ t |

< IA o i + IA11 • • • 4- i A

| + A , - _ i | -+■ jA a | .

Suppose (i'E)EeJr is a solution of (3.18). We next show it is ecjual to the all-zero vector. By Lemma 3.2.4. we only need to show that l s — 1 . We distinguish two subcases:

Case 2.1. 1(Eq) = .s — 1. in this case it is obvious that (.’£•„ = 0. since x/r0 =

0

is

one of the linear equations in (3.18). Case 2.2. l{En) > s. let / be such th at I < E q and / G A*,. N ote th a t if £ () 6 A’,, we can take / to be £(). We contend th at L / = x e q■ Otherwise there exists F € Jwith F / f

0

and / < F. This implies th at I < F A

£ 0

and l ( F A £,,) > .s. which

forces /(£ A £ 0) = ls. But this is a contradiction because 1{EQ) < ls. Now (c/r) satisfies L[ =

0

or equivalently xe 0 =

0

. so ve0 =

0

.

□

L e m m a 3.2.8. Let (A*. < ) be a quasi-poly normal semi-lattice. I f J- isan L-intersection family for L = {0 . 1 . • • • . s — 1 }. then \E\ = |A'0| + |A 'i| ■■■+ |A 'S| if and only if T

= A'0 U A'i • • •U A '.,.

P ro o f. The sufficiency is obvious. Next let us prove its necessity. Let I = | U - _ 0 .V, | = consider the |JF| by I incidence m atrix M whose rows are indexed by

IZt=o

elements of J£ and whose columns are indexed by elements of Ys. where Y' := U-_0 A',. For .1

6

£ 5

e Ys, the (.4,5)-entry of M is defined to be 1 if either A G U and

5 = A or A G A' — Y's, S G .Ys and S < A. It is defined to be 0 otherwise. 50

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Observations: It is clear from the above definition of M that ( 1) if the (.-I. S)-entry of M is 1 . then 5 < .4. ( 2 ) each row has at least one non-zero entry, and (3) ifA

6

i f and A € A'„. u > s. then the row corresponding to .-I has f s{mu) >

1

non-zeroentries ( see C hapter 2 §1 for the definition of quasi-polynomial semi-lattice). Claim: For A ^ B € T . the (.4. £?)-entry in M M r is 0. Proof of the claim: Suppose (.4. B )-entry of .1 /4 /r is > 1. Then there exists an S’ € Y, such th a t both the (,4.5)-entrv and the (B .S )-e n try of 4 / are (1). S < A A D. so S € U-TqA*, = F,_i since JF is a {0.1 family.

1

. By observation

s — I {-intersection

But from the definition of 4 /. for such an S. the (.4. S)-entry of M is L if

and only if .4 = 5 . The same is true for the (B . S)-entry. So .4 = S’ = B. which is a contradiction. This proves the claim. From the above claim, it is clear th at M M r is a diagonal m atrix, and it is also clear th at the diagonal entries are non-zero by observation (2) above. So M M 1 is a non-singular |JF| by |JF| matrix. Since |JF| =

|A'j|. M is a square m atrix. It is clear that each row of 4 / can

contain at most one non-zero entry, otherwise M M T would not be a diagonal matrix. So the total num ber of Fs in M is
.s. lemma 3.2.7

completes the proof. If L is not of the form {0. I . - - - ..s — 2./,} for some /s > s. lemma 3.2.5 completes the proof. Case 2. L is consecutive but not equal to {0. 1. • • • . .s — 1 }. In this case. F has no gap element and therefore by lemma 3.2.4 we know that the system of linear equations: for all I G A., U A s-.> U A , j ■• • A s_>[., has only the trivial solution. From this we deduce th a t the number of variables is less than or equal to the number of equations, i.e. F\ < |A ',| + |A ,_ 2| + |A',_4| + • • • |A.,_•»(.,/oj

which completes the proof of Theorem 3.2.1.

52

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CHAPTER 4 MODULAR COMBINATORIAL INEQUALITIES

In this chapter, we study some m odular combinatorial inequalities. First we generalize Frankl and Wilson's modular version of the Ray-Chaudhuri-W ilson Inequality to a quasi-polynomial semi-lattice, and then prove a theorem which supports a conjecture by Alon. Babai and Suzuki to a large extent.

4.1

Some Basic Modular Combinatorial Inequalities

In this section, we generalize Frankl and Wilson's m odular version of the RayChaudhuri-W ilson Inequality [1 1 ] to a quasi-polvnomial semi-lattices. T he proof is essentially from that by Frankl and Wilson [11]. First we introduce some notation. Let p be a prime num ber. Let (A’. < ) be a quasipolvnomial semi-lattice with height function !(.), L be a subset of {0 . 1 . ■• • .p - I}. We call F with F C A’ a mod p L-intersection family if |E A F | any E. F

6

F with E

6

L

(m od p) for

F. Here for an integer m and a set A of integers, rn € A

(mod p) means that there exists an integer a

6

.4. w ith m = a (mod p). Finally for

any m atrix M over Z we use r a n k ( M ) to denote the rank of m atrix M over Q and

53

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we use rankp(M) to denote the rank of the m atrix M over Fp (abbreviated in this chapter as F), the finite field with p elements. Recall th a t any polynomial sem i-lattice (A'. . ■■■. m0)(x —mi) • • • (x —

For any i = 0. 1.2.

. s. we define gt(x) --

-

It is clear th at there exist s 4-1 integers />0. b{. ■■• . 6 , with

(x - li)(x - l2) ■• • (x - ls) =

biPiix) (mod p). So (mod p).

where di{= b{C,) is an integer for i =

0

, 1 . 2 . • • • . s. since by the definition of a quasi­

polynomial sem i-lattice in C hapter 2. f t(x) = (x - m 0)(x — mi) ■• • (.r - m t^ i ) / c l and e.i is an integer for i =

0

. 1 . ••• . s —1 .

For any two integers u and u w ith 0 < u < v < n we define I(u. v) to be a

0-1

incidence m atrix whose rows and columns are indexed by the elements of A'u and A',, respectively.

For A € X u. B € A’„. the (.4., B )-entrv of I(u, v) is

1

if .4 < B and

0

otherwise. For any integer i with 0 < i < s and .4 € AT B

6

AT, the (.4. £?)-entry of

I ( i, s ) I { s ,k ) is easily seen to be the number of elements 5 with 5

6

A', and .4


bv fact

2

above

'

67

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by taking i = 0 in lemma 4. which completes the proof of the theorem.

68

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CHAPTER 5 MISCELLANEOUS RESULTS

In this chapter we give some miscellaneous results on C om binatorial Inequalities. The first section is on Snevily’s conjecture, where we give proofs of his conjecture in some special cases. T he second section is on the extreme case of the Alon-Babai-Suzuki type inequality. In th at section we will show that there is only one family that satisfies the bound given in the the Alon-Babai-Suzuki type inequality under a mild condition.

5.1

Special Cases of Snevily’s Conjecture

In [2-1] Snevily m ade the following conjecture.

Conjecture 5.1.1. (Snevily) Let Y be a set with n elements, T be a family of subsets of )' and L C I n be a set with s elements. If (F is an L-intersection family with I >

0

.

V/ € L. then \E\ < E U (n; 1)The main results of this section are the following theorem s which confirm Snevily's conjecture when s = 2 and in some special cases when s > 2.

Theorem 5.1.1. Snevily's conjecture holds if s = 2. Before we give the proof, let us recall that E is called a gap element with respect to L = {/t. l2, ■• ■ , /s } if E

6

T and r ii= i( l^ 'l—k) < 0- S° in particular, if L = {/[. /•>} 69

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with l\ < In. then E G T is called a gap element if lx < \E\ < In: if L = with l[ < In < 7?. then E G T is called a gap element if In < |E | < hP ro o f. Let L = {/[./•>} with

< In. As in Chapter 3. for each E € F we have a

variable x e and we define a linear form for each / C l ' as follows: Li -

Xe~ i c e

. E e r

We consider the following system of linear equations:

If one could show th a t this system of linear equations has the trivial solution 0 (the all-zero solution) only, then by linear algebra T | = number of variables in (5.1) < ip 2( n u p 0( n i

and so this would prove the theorem. Let ( u e ) e ^x' ( or ( u e ) for short) be a solution to the above system of linear equations. It is enough to show th a t ile = 0. VE G T . By lemma 3.2.4. it suffices to show that ue = 0 for all gap elem ents E G T . Recall that in this situation. E is called a gap elem ent if ^ < \E\ < In. O f course if In = l\ + I. then there is no gap element. So let us assume that l2 > k 4- 1. Let E0 be a gap element. We note th a t |£o| > h + l > 1 + 1 = ‘2. To show th a t ue0 = 0 and complete this proof, we distinguish two cases. 70

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Case 1. /[ = 1. We need a sim ple but useful lemma.

Lemma 5.1.1. Let L = {/[, L, • • • . ls\ and F be an L-intersection family. If there ; .........................

fcO t v t b 6/ H/Cty L/

live

^r

^ A. «i*«

■• ,

i

o ,1 1 . . ^ , ^ _c Uflilii

UH

A/

-r

*C •/

U y66/ t>

i

^

I iri

^

-v.

I

I

tl—

r

L* I

_

..

— a fc.

for any I C Y with lu < |/ | < /u+i and I C E . P ro o f. By definition L[ = Y I f z f

i c f x f

-

^

enough to prove th at there is no

F € F with I C F and F ^ E. Suppose there is F G F w ith I C F and F

E.

then I C. F n E and so /„ < |/| < \F fl E\ < |E | < /II+1. This implies \F fi E\ L. a contradiction to the fact that F is an L-intersection family. This proves the lemma.

□

.Now we return to Case L. Let I be any subset of £ 0 with |/ | = 2. By lemma 5.L.1 L[ = x Eo. Specializing x E = aE. V£ 6 IF. we have 0 = L[{{uE)) = aEt) since L / = 0 is one of the linear equations in (5.1). This completes the first case. Case 2. 11 > 2 . We have the following identity:

In fact it is clear th a t both sides are linear combinations of x / r 's where F 6 F . For any F € F . the coefficient of xp contributed by the second term in the left hand side is equal to the num ber of I €

such th at / C

£ 0

and I C F

71

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by the definition of L[. This number is easily seen to be (|Fn,Foi). So the coefficient of ./-/•' in the left hand side is equal to (|Fn,£'0^) — ((,1) by the definition of L$. By the fact th at /[ < |E 0| < /•> and the fact th at E is an Z-intersection family, we know th at for any F G E . \ F n E 0\ = L if F ^ E0. | E n E 0| = |E 0| if F = E (). So for any F € E . ( !F

7

0i) -

(2)

=

0

if F

^

E « and

( | Fn/ 0i ) -

(2)

=

0

2°') -

(2)

if F

=

E «>- T U i y P r o v e s

(5.2).

Since ( u e ) satisfies Li — 0 for any / € P-^V) U P0(1'). we have ( ( |/:,°l) — ((j ))«£•„ = 0 by specializing x e = »£■ for all E € E in equation (5.2). (IE,)') _

q an[j SQ wp ^ave

_ q g y jemma

3

2 .4

It is obvious th at

this completes the proof

of the theorem.

□

The following theorem which states th a t theorem 5.1.1 holds in the more gen­ eral context of quasi-polynomial semi lattice. The proof follows from the above one verbatim.

Theorem 5.1.2. Let (X . 3.

72

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P ro o f.

The proof is sim ilar to that of theorem 5.1.1. We consider the system of

linear equations |L / = 0 Lot {".g)

/

6

P i( l') U P:t(F).

a «o!ution. wp need to 15how th a t >!g =

If /,( = 3. then there is

110

0

for all E

E.

gap element and so by lemma 3.2.4 (iig) is the trivial

solution. Now suppose th a t Z3 > 3 and there is a gap element Eq € J- with 2 < !£j)| < /■(. By lem m a 3.1.1. we have the following claim. For any fixed I C E{) with |/ | = 3 L[ = x g 0. Specializing x g = ug, V E

6

T , we have

0

= L[((ug)) = ugQ =

0

and by lemma 3.2.4.

(ug) is the trivial solution. This completes the proof of the theorem.

□

The m ethod used in theorem 5.1.1 can also be used to prove th at Snevily's con­ jecture holds in another two cases. For their proofs, we consider the following system of linear equations. f L*/2J - h + 1. l->+ 2. • • • . l->+ s —'2\. P r o o f . Suppose (uE) is an arbitrary solution of (5.3). It is enough to show that (u E ) is the all-zero solution. By lemma 3.2.4 it suffices to show th at uE = 0 for all gap elements E. If s is an odd integer, then there is no gap element for L in this situation. So in the following, we assume th a t s is even and E is a gap element and distinguish two cases to prove uEa =

0

. Notice th a t for any / € P_>(1’). L[ = l) is a

linear equation in (5.3) if s is even. Case 1.

=

1

.

In this case. \E\ > /i -I-1 > 2. For any I C Y with |/| = 2 and / C E. L[ = x E by lemma

5.1.1.

Now if we specialize x E = uE, ViT

L[ = 0 is a linear equation in

(5.3).

6

E , then 0 = L[((ile )) = uEt] since

This proves th at uE = 0 in this case.

Case 2. /[ > 2. Just like equation (5.2). we have the following equation

teP 2(V). icEo

Now if we specialize x E = uE, VE

6

E , then th e right hand side of the above identity

is zero and so ( ( ^ ol) — 5))ue 0 = 0. Therefore u Eo = 0 since (!^oi) — (!j) ^ 0.

74

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Now by lemma 3.2.4 this implies that (ug) is the trivial solution and hence the proof of this theorem is complete.

□

The following case of Snevily's conjecture requires a quite different proof. T h eo rem 5. 1. 6. S r)plnl'J

Proof.

rnnjprhirp hnhls f n r

= fl ,3,5V

Just like the proof of theorem 5.1.1. we need to show th at the following

system of linear equations

L ,= 0

/€

P ^ n u lP atr).

(o.4)

has the trivial solution only, since that would imply \F\ = num ber of variables in (5.4) < rank of linear equations in (5.4)

< |P i(V )U P 3(J')| n\

(n

l) + l3 n o

1

\ M

(n -

1

\

(n -l\

1 ) +(

2

M

(n -

1

3

and thus prove the theorem. We suppose ( u e ) e €J: is a solution to equation (5.4). By lemma 3.2.4. in order to show(aE) is the trivial solution, it suffices E

6

to show th at uE =

0

for any gap element

T . Recall th a t by definition, E G T is a gap element if 3 < |£ j < 5. If there is

no such gap element, then the lemma 3.2.4 applies. So we suppose th at there is an element E. w ith 3 < |£7| < 5 i.e. |£ j = 4.

Io

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For any / 0 C E w ith j/01 = 2. we have the following identity

L [0 =

^2

( 5 .5 )

L [ —xe .

| / | = 3 , I qC I C E

In fact, it is clear th a t both sides are linear combinations of xp's where F € J-. If [0 U.

□ Now let us re tu rn to the proof of Theorem 5.2.2. We need a series of lemmas. Recall that for a fam ily T of subsets of 1 \ we define a linear form L\ - Y ^ E c f

r i-: *;or

any I C Y. Here x/r is a variable associated with E. VE G IF. L e m m a 5.2.2. Let b. a be two positive integers with b < a. u t . u->. - - ■ . n„ be nonnegative integers with tii < u-> < ■■■< ua and K = {A:[. k>. ■■■. k(,} with

< k> < ■■■
. ■■■ .kb) shadows (ua_b+i- ua_b->. • • • - O - then any solution of

I l[ = 0

vi €

I

\J pUt( r )

(5.10)

i=u-6 +l

is also a solution of

|^ = 0

V / g j j F J V ’).

(5.11)

P r o o f of le m m a 5 .2 .2 . It is sufficient to show that for any J € U?=o

).

Lj

is a linear com bination of L r's with I running through U?=a- 6 +i Suppose | J\ = uj w ith j < a — b. Consider the following (c. r/)-entrv is (V u a _ke~uJ ) where c.d = 1,2, • • • . 6 . \ i . 6 + tl — uj!

6

x

6

m atrix M whose

Since \( k i . k - u - - - . k bw) shadows

{ua-b+i, aa-b+2 - • • • • ua)r we have (Art — uj, k -2 — Uj,- ■■ , k b ~ Uj) shadows (ua_ ^ [ — 84

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Uj. ua-b+ 2 ~ Uj- • • • • ua ~ uj)- By theorem 5.2.3 we know th a t M is not singular. So Ca—6+1 1

^a—6+2

there exists v =

i.e. we have

with M v =

w y

t\i(^c

Uj\

= 1

for all c =

1

. 2 . • • • . b.

5 . 12 )

We claim th at y £=0-6+1

^

J 2

L/ =

L j-

/ePu,(K), J C I

In fact, it is clear th a t both sides are linear combinations of x/Ts where F € T . For any F € T with F 2 J- the coefficients of x p in both sides are easily seen to be

0

. If

F € T with F D ./. the coefficient of x/r in the left hand side is equal to c, • (the number of / € PK|(1") such that J C / C F).

5 .1 3 )

:=a-6+1 It is clear that (the num ber of / € P u,( i ) such th a t J C / C F) =

|F! - uj u, — uj

So (5.13) is equal to X^Lo- 6 +i ui ( P - u 1) which is equal to 1 by (5.12). This completes the proof of the lemma.

□

The sufficiency of the condition given in the statem ent of theorem 5.2.2 is obvious. O ur approach to prove the necessity is to prove th a t if L ^ {0,1. • • ■..s — 1}. then the upper bound can not be achieved. This is done in the following lemma. 85

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L e m m a 5 .2 .3 . 1

}.

W ith

th e s a m e

h y p o th e s is

a s in

th e

5.2.2.

I f L

^

{0. 1. • • • . s —

th e n

|J:| ki 4- i — I. Since k i > .s - r + I. we have k{ ^ k [

1 —1

>,s - r +

+ (-

1

1

= s — r 4- i

So

{ k \ . k- > .

••• . k

r )

shadows

(s

= s

-

> s

—2 r 4- 2 i .

2

r -f

2 i

4- (r —i)

— 2r 4- 2. s — 2r 4- 4. • • • . s ) . By the lemma 5.2.2. any

solution of | l

/ =

0

V / 6 PJ( r ) u P , . 2 ( K ) U " - U P , - 2r+2( n

(5.14)

is also a solution of the following: I/=0

V /€ P .,( y ) U iP i _.2 ( r ) u ! P s_.1 ( V' ) U- - -

By lemma 3.2.4. we know (5.15) has the trivial solution only.

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(5.15)

This implies th a t |JF|
. ■■• . A;r_i} with k[ < k> < ■■• < A-r-iIn the remaining part of the proof of the lemma, let us assume that L [ is redefined as

Since (&[.£>.•• • .Av_ 1 ) shadows (.s —r + l . s - r + 2. • • • .s - 2 ..s ) . by lem m a 5.2.2. any solution of

is also a solution of the following system of linear equations V/

€ PS(V) U P S- , ( F ) U Pa_3( l') U • • • U Po(l').

Let (up) be a solution to (5.21). Fix such a gap element E 6 P w ith a — 2 < |£ |
s. Let I be a subset of E with |/| = s. By lemma

5.1.1 Li = x E- Specializing x E =

for all F £ T ' , we have L[{(uE)) = u E = 0

since L / = 0 is one of the linear equations in (5.21). Now Lemma 3.2.4 implies th at

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(up) has to be the trivial solution. So (5.21) has th e trivial solution only and so does (5.20). We have \E'\
\

/

n

\

/

n

\

/

n

\

~ ( a j + {« - 2 j + ( S - 3 j + " ' + ( , ' - r + j \F \ = \ F \ + \ F n r

m d S°

,_,(}•)!

This completes the proof of case 3.3 and th a t of lemma 5.2.3.

□ L e m m a 5.2.4. We make the same assumption as in the theorem. If L = {0.1. • • • . .s— 1}. then \T\ = (") 4- ( , " L) H

+ C_r+i) lf and oniy lf ? ~

U P.,-i U --

P r o o f o f L e m m a 5 .2 .4 . The sufficiency is obvious, so let us prove the necessity. Construct a 0-1 m atrix M whose rows and colum ns are indexed by elements of T and P ,_r+i U P , _

r +2

U

■• ■U Pj respectively.

For any E 6 IF and .4. € P.,-r+i U P 5 defined to be 1 if E = Observations: It is

r + 2

U • • • U P „ the (E . .4)-entry of M is

A with |£ | < s — 1 or A C E

with |.4| = s and 0otherwise.

clear from the above definition of M th at

(1) if the (E. .4)-entry of M is 1, then .4 C E ,

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(2) each row has at least one non-zero entry since k > s - r + 1. VA: 6 A’ by the hypotheses. (3) if E € T with \E\ > s. then the row corresponding to F has ('*/) (which is greater than i) nun-zero entries. Claim: For E # F € F . the (E . F)-entry in 4 / 4 / T is 0. Proof of the claim: Suppose (F , F)-entry of 4 / 4 / r is > I.

Then there exists an

.4 C Y with s - r + 1 < |.4| < .s such that both (E . -4)-entry and (F. .4)-entry of 4 / are L. By observation (1). .4 C E O F . so .4 6 U'JTqP^}’) since F is a {0. 1........* - 1}intersection family. But from the definition of 4 /. for such an .4. (E. ,4)-entrv of M is 1 if and only if E = .4.

The same is true for the (F. .4)-entry. So E = .4 = F .

which is a contradiction. This proves the claim. From the above claim, it is clear that \ I M T is a diagonal m atrix, and it is also clear th a t the diagonal entries are non-zero by observation (2) above. Since \F\ = ]T^=s-r+i ("). M is a square m atrix. We observe th a t each column of M can contain at most one non-zero entry, since otherwise 4 / 4 / r would not be a diagonal m atrix.

So the to tal number of l rs in 4 / is < X ^-s-r-^i (”) = |F |.

observation (2) the total num ber of l ’s in M is > |F |.

By

So the to tal number of

l ‘s in XI is exactly |F |, i.e. each row of XI should contain exactly one non-zero entry. By observation (3), this means that E € U i=s-r+i

) f°r

an>' E € F . So

^ C U L .- r + l P .(V ). But 1^1 = £ ; = . - r +i (*), so r = U;=I. r+1P ,( l') .

Now the proof of theorem 5.2.3 is easy.

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□

P r o o f o f T h e o re m 5 .2 .3 .

The sufficiency is straightforw ard.

necessity. Suppose \F\ =

Let us prove the

(")• By lemma 5.2.3. L = {0. 1. • • • . .s - 1}. Then

by lem m a 5.2.4. F = P., U P,._i U • • • U P.,_r+i. This proves the theorem 5.2.3.

W ith a similar proof as above, one can prove th a t the same result holds m the vector space case. Recall th at a family F of vector subspaces is an /.-intersection family if for any E, F 6 F with E

F. d im (E D F) € L.

T h e o r e m 5 .2 .4 . Let n. r. s be three positive integers with s
s — r + 1. VA: 6 K .

If a family F of

vector subspaces is an L-intersection family with d im ( E ) € K . ' i E € F . then \F\ < H «=.s-r-i[rl- anti further \F\ = I] -_ ,_ r+l["] if and only if L = {0. 1. • • • . .s - 1} and ? = U L , - r + [ Pt(V). where P ,(V) for a vector space 1’ is the family of all subspaces of dimension 1 .

□

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BIBLIOGRAPHY [L] X. Alon. L. Babai. H. Suzuki. "Multilinear Polynomials and Frankl-RayChaudhuri-W ilson Type Intersection and Theorems." J. Combinatorial Theory (A) 58(1991) 165-180. [2] I. Anderson. Combinatorics of Finite Sets. Clarendon Press. Oxford 1987. [3] L. Babai. P. Frankl. "Linear Algebra Methods in Combinatorics" D epartm ent of C om puter Science. University of Chicago, 1992. [4] K. Borsuk. "Drei Siitze iiber die n-dimensionale euklidische Sphare" .Fanil. Math. 20 (1933) [5] R. C. Bose. "A note on Fisher’s inequality for balanced incomplete block de­ sign". Ann. M ath. S tat. 20 (1949) 619-620. [6] X. G. de Bruijn and P. Erdos. " On a Com binatorial problem." Proc. Kon. Ned. Akad. v. Wetensch 51(1948) 1277-1279. [7] P. Frankl. "Intersection Theorems and Mod p Rank Inclusion Matrices" ./. Combinatorial Theory (A) 54(1990) 85-94. [8] M.Deza. P.Frankl and X.M.Singhi. "On functions of strength t.” Combinntorica 3 (1983) 331-339. [9] P. Frankl and Z. Fiiredi, "Families of Finite sets with missing intersections." Colloquia Mathematica Societatis Janos Bolyai 37:305-320. 1981. [10] P. Frankl and R. W. W ilson, "The Erdos-I