Studies in Pure Mathematics: Papers in Combinatorial Theory, Analysis, Geometry, Algebra and the Theory of Numbers, presented to Richard Rado on the occasion of his sixty-fifth birthday 0124984509

Table of contents :
Title
Preface
Contents
Reinhold Baer: The Determination of Groups by their Groups of Automorphisms
R. A. Brualdi, J. S. Pym: A General Linking Theorem in Directed Graphs
N. G. de Bruijn: The Exterior Cycle Index of a Permutation Group
H. S. M. Coxeter: The Finite Inversive Plane with Four Points on each Circle
H. G. Eggleston: Intersections of Open Plane Sets
P. D. T. A. Elliott, H. Halberstam: The Least Prime in an Arithmetic Progression
P. Erdös, A. Hajnal, E. C. Milner: Polarized Partition Relations for Ordinal Numbers
P. Erdös, A. Méir, Vera T. Sós, P. Turán: On Some Applications of Graph Theory, II
T. Estermann: Notes on Landau’s Proof of Picard’s ‘Great’ Theorem
D. R. Fulkerson: Disjoint Common Partial Transversals of Two Families of Sets
Marshall Hall, Jr.: Affine Generalized Quadrilaterals
H. Heilbronn: On the 2-Classgroup of Cubic Fields
Edmund Hlawka: Discrepancy and Riemann Integration
Alan J. Hoffman: On Vertices near a Vertex of a Graph
H. A. Jung: Connectivity in Infinite Graphs
N. S. Mendelsohn: Intersection Numbers of t-Designs
L. Mirsky: A Proof of Rado’s Theorem on Independent Transversals
C. StJ. A. Nash-Williams: Edge-Disjoint Hamiltonian Circuits in Graphs with Vertices of Large Valency
B. H. Neumann: Algebraically Closed Semigroups
A. Oppenheim: The Irrationality or Rationality of Certain Infinite Series
Hazel Perfect: Marginal Elements in Transversal Theory
Gian-Carlo Rota: On the Combinatorics of the Euler Characteristic
Olga Taussky: Some Remarks on the Matrix Operator ad A
W. T. Tutte: Graphs on Spheres
H. Tverberg: On Equal Unions of Sets
B. L. van der Waerden: How the Proof of Baudet’s Conjecture was Found
D. J. A. Welsh: Related Classes of Set Functions
Publications of Richard Rado
List of Contributors
Back Cover

Citation preview

Studies in Pure Mathematics

Papers presented to

Richard Rado

This volume is offered as a tribute to the achievement of Richard Rado in many branches of pure mathematics, and the choice of authors and papers reflects the wide spectrum of his own investigations. Many contributions are devoted to aspects of combinatorial mathematics, the area in which Rado has carried out his most intensive researches and in which he is recognised as the leading authority. Particular stress has been placed on such topics as the

theory of graphs, enumerative analysis, the partition calculus, transversal theory, and combinatorial designs. This book is among the few works, in any language, dealing largely with combinatorial

mathematics. The broad range of Rado's work is also acknowledged in the papers contained in this book which are divided between analysis. geometry, algebra and the theory of numbers.

It is indeed possible that “Studies in Pure Mathematics" will become a classic to all mathematicians.

£5.00/1005-

STUDIES IN PURE MATHEMATICS

STUDIES IN PURE MATHEMATICS PAPERS IN COMBINATORIAL THEORY, ANALYSIS, GEOMETRY, ALGEBRA, AND THE THEORY OF NUMBERS

presented to

RICHARD RADO on the occasion ofhis sixty-fifth birthday

edited by

L. MIRSKY

1971

Academic Press: London and New York

ACADEMIC PRESS INC. (LONDON) LTD Berkeley Square House Berkeley Square London, WIX 6BA

US. Edition published by ACADEMIC PRESS INC. 111 Fifth Avenue New York, New York 10003

Copyright © 1971 By Academic Press Inc. (London) Ltd

All Rights Reserved No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without permission from the publishers Library of Congress Catalog Card Number: 78-129790

ISBN: 0-12-498450—9

Printed in Great Britain by ROYSTAN PRINTERS LIMITED

Preface

On 28 April 1971 Richard Rado, Professor of Pure Mathematics in the University of Reading, celebrates his sixty-fifth birthday; and a number of

his friends and colleagues who share his interests welcome this opportunity to pay tribute to a remarkable mathematician and a man held in universal regard. In an age when specialization has become the norm, Rado’s long series of imaginative and highly individual discoveries testifies to a degree of versatility that is altogether out of the common. Classical analysis and the study of inequalities, the theory of numbers and algebra, geometry and the exploration of convexity, measure theory and transfinite arithmetic have all been grist to his mill. Yet his deepest insight has been displayed in combinatorial mathematics, and it is as the leading figure in this field of research that he is most widely known. To gauge the magnitude of Rado’s contribution to com-

binatorial studies, it suffices to recall the selection principle now linked with his name, his work on ‘regular’ systems, on linear combinatorial topology, on the theory of graphs, on independence structures and independent trans-

versals, and above all his extensions of Ramsay’s theorem and the resulting development of the ‘partition calculus’.

The papers oflered here are intended to reflect both aspects of Rado’s work: its broad range as well as the dominance of combinatorial questions. The authors wish Richard Rado many further years of fruitful mathematical activity, and they are confident that their good wishes will be echoed and their admiration for his achievement shared by a much wider circle of mathematicians than that formally associated with the present volume. University of Sheflield January, 1971

L. MIRSKY

Contents REINHOLD BAER The Determination of Groups by their Groups of Automorphisms

R. A. BRUALDI and J. S. PYM A General Linking Theorem in Directed Graphs

N. G. de BRUIJN

17 The Exterior Cycle Index of a Permutation Group

H. S. M. COXETER on each Circle H. G. EGGLESTON

31

The Finite Inversive Plane with Four Points

39 Intersections of Open Plane Sets

P. D. T. A. ELLIOTT and H. HALBERSTAM an Arithmetic Progression

The Least Prime in

P. ERDOS, A. HAJNAL, and E. C. MILNER Relations for Ordinal Numbers

Polarized Partition

53 59 63

P. ERDOS, A. MEIR, VERA T. SOS, and P. TURAN On Some Applications of Graph Theory, 11

T. ESTERMANN Theorem D.

Notes on Landau’s Proof of Picard’s ‘Great’

101

R. FULKERSON Disjoint Common Partial Transversals of Two Families of Sets

MARSHALL HALL, JR. H. HEILBRONN

Afiine Generalized Quadrilaterals

On the 2-Classgroup of Cubic Fields

EDMUND HLAWKA ALAN J. HOFFMAN H. A. JUNG

89

Discrepancy and Riemann Integration On Vertices near a Vertex of a Graph

Connectivity in Infinite Graphs

N. S. MENDELSOHN

Intersection Numbers of t-Desigus

107 113 117 121 131 137 145

L. MIRSKY A Proof of Rado’s Theorem on Independent Trans-

151

versals vii

CONTENTS

C. St.J. A. NASH-WILLIAMS

Edge-Disjoint Hamiltonian Circuits

in Graphs with Vertices of Large Valency B. H. NEUMANN

Algebraically Closed Semigroups

Marginal Elements in Transversal Theory

GIAN-CARLO ROTA teristic OLGA TAUSSKY

203

On the Combinatorics of the Euler Charac-

221

Some Remarks on the Matrix Operator ad A

Graphs on Spheres

On Equal Unions of Sets

B. L. van der WAERDEN was Found D. J. A. WELSH

185 195

HAZEL PERFECT

H. TVERBERG

157

The Irrationality or Rationality of Certain Infinite

A. OPPENHEIM Series

W. T. TUTTE

viii

235 239 249

How the Proof of Baudet’s Conjecture

Related Classes of Set Functions

251 261

PUBLICATIONS 0F RICHARD RADO

271

LIST OF CONTRIBUTORS

275

The Determination of Groups by their Groups of Automorphisms REINHOLD BAER

It has been noticed quite often that restrictions, imposed upon the group of

automorphisms, lead to even stronger restrictions of the structure of the underlying group. It is the objective of the present note to offer a few further instances of this phenomenon. If m is some class of groups, then one may define a class (0* of groups by the property: G e (0* if, and only if, Aut F e co for every factor [= epimorphic image of a subgroup] F of G.

In Chapter 1 we investigate the class (0*, in case a) is the class of abelian groups or the class of countable groups or that of locally finite groups.

Chapters 2 and 3 are devoted to a characterization of the class of all almost cyclic groups [= groups with a cyclic subgroup of infinite index]. Generalizing Plotkin’s class of radical groups we term the group G an Ell-group if every epimorphic image, not 1, of G possesses an accessible subgroup, not 1, which is locally finite or locally nilpotent. Our principal result, obtained in

Chapter 3, may now be stated as follows: The class of almost cyclic groups is the most comprehensive class R of groups with the following two properties: Factors and groups of automorphisms of R-groups are R-groups; and every R-group, not 1, possesses an accessible locally-SR-subgroup, not 1. An important step in the proof of this result is constituted by the proof of the fact that every locally-almost-cyclic group G possesses a characteristic

locally finite subgroup L such that G/L is either 1 or a torsionfree abelian group of rank 1 or a non-abelian extension of a torsionfree abelian group of rank 1 by a group of order 2. Notations Aut G = group of all automorphisms of the group G cXY = centralizer of Y in X 36 = center of G

A 0 and a component K of 630) such that the frontier ofK is afinite number ofdisjoint simple closed Jordan curves.

For any a > 0, Gs“) is an open set and, if (g denotes complements, g(GS(a)) =

U S(ps a)' 116‘616)

Thus each point of Fr(Gs(,,)) lies on a disc of radius a that does not meet

GS“). Let K 1 be a component of Gs“). Since each component of ‘6’(K1)contains an open disc of radius a there are only finitely many of them (there is

INTERSECTIONS OF OPEN PLANE SETS

55

only one unbounded component). Of these components some may be at a

positive distance apart. If there are any such, let their least distance apart be 36. If there are no components whose closures intersect, we write K2 for K1. If there are such components and 6 is defined we write K2 for K150). If there are such components and 6 is not defined, we write K2 for K150”, where n is so small that K150” ¢ 9. Take r to be a, a + 6 or a + n as the case may be and

K to be K2, or a component of K2. Then each component of Fr K is of finite length, locally connected and hence

arcwise connected. Each component bounds two domains in the plane (for the components of ‘fiK are at a positive distance apart). Hence each component of Fr K contains a simple closed Jordan curve.

But each component of Fr K is a simple closed Jordan curve. For otherwise we can suppose that a component L of Fr K contains a point p which lies on three distinct arcs r, a, p terminating at p and on the frontier of S(p, 6) for some 6 > 0. (Fig. 1). Moreover I, a, p will not meet except at p. They divide

the interior of S(p, 6) into three domains say A, B, C. Of these two at least contain points of 93K arbitrarily close to p. Since these points lie on closed discs lying in @K and of radius r, we see that one of the arcs 1:, a, p separates two such discs as in the figure below. Let the centres of these discs be a, b.

S(p, 5)

FIGURE 1

The two discs must be in the same component of @K since distinct components are at a positive distance apart. We can thus join a, b in WKK by a polygonal line and adding segments pb, pa to this polygonal line obtain a connected subset of ‘KK that separates points of ‘L' from points p and 0'. But there are points of K arbitrarily close to these points of 1." and to these points of p, a. As K is connected, this is impossible. Thus the whole of L is a simple closed Jordan curve.

H. G. EGGLESTON

56

LEMMA 2. D is a domain whosefrontier is a simple closed Jordan curve. t is apoint of the segment pq such that t¢ D, p e D, qe D. For any point u not on the line pq, B(u) denotes the open half-space not containing p or q and bounded by the line parallel to pq through 1;. Then there exists a point u on Fr D such that, for some n > 0, S(u, 11) n B(u) E D.

(6(5) meets line pq in a relatively open set of which one interval contains t. Let the end-points of this interval be p1, ql. There are two arcs of Fr D with end points at p1, q1 say a1, :12 which bound with the segment p1 q1 domains denoted by D1, D2 respectively. One of these domains contains the other, say D, 3 D2. Of all the points of «2 choose one, say u, that is furthest from the line pq. Then, for any 6 > 0, S(u,6) n B(u) n 012 = D.

u is at a positive distance from 0:1 ; thus for 6 sufliciently small, say 6 < r], S(u,6) n B(u) n (11 = Q.

Also S(u, n) n B(u) can be joined to arbitarily distant points by an arc that does not meet 0:2 or the segment p1 ql. Thus

S(u, n) n B(u) E ‘€(Dz)~ Since ueoz2 E D1, we have S(u, n) n B(u) c 0,. Thus S(u,n) n B(u) 9. D1 — Dz = D. We can now establish Green's conjecture for the plane case. Given the open bounded set G construct, as above, a set of vectors X such that one component K of GX satisfies Lemma 1. Let H be the convex cover of K . For p e FrH let T(p) denote the intersections of those open half spaces bounded by support lines to H through 1) which contain H. Next select three arcs of Fr H, at, [3, y, where each is either an arc of Fr K or

is a segment that meets Fr K only at its end points. Denote by T(a) the set fl T(p), where (1° is the interior of a relative to Fr H, and define T(fi), T(y) peao

similarly. We can, and will, choose at, [3, y, so that T(a) n T(fl) n T(y) is a bounded set.

If a, [3, y are all arcs of Fr K, select points a, b, c, of a0, [30, y°, respectively such that T(a) n T(b) n T(c) is bounded. Then H n H(a—b) n H(a—c)=@.

Now for 0 < A < l, H n H(l(a — b)) n H(l(a — c)) is an open convex set. If A is sufficiently near to 1

H n H(Ma — b)) nH(l(a — c)) = K n K(}.(a — b)) n K(l(a — c)).

INTERSECTIONS 0F OPEN PLANE SETS

57

Thus we have a set of vectors X such that Kx is an open convex set. If one of the arcs a, [3, )2, say a, is not an arc of Fr K we apply Lemma 2. Let X be the set of vectors of the form la, where 0 < A < 6, 6 is a fixed positive number and

a is a unit vector parallel to on. Then KX contains a component K1 whose frontier contains a segment parallel to a. If 6 is sufficiently small and H1 is the

convex cover of K1, Fr H1, contains arcs that are translations of parts of [3, y say A, 'yl. If [31, yl are arcs of Fr K1, we can now argue with 0:1, B1, yl as we did with a, B, )2 above. We find X 1 a set of vectors such that K 1 x1 contains a convex component. By the remarks made earlier, this is sufficient. If one of [31, 3),, is not an arc of Fr K‘, we apply Lemma 2 again, and if necessary apply it once more. In any case we are led to an open convex set, and Green’s conjecture is proved. Reference 1. GREEN, JOHN, W. On families of sets closed with respect to products, translations and point reflections. Anais Acad. Brasil. Ci. 24 (1952), 241-244.

The Least Prime in an Arithmetic Progression P. D. T. A. ELLIOTT AND H. HALBERSTAM Let P(q, h) denote the least prime in the arithmetic progression h mod q, (h, q) = l, l < h < q, 2 < 1]. In 1934, S. Chowla (l) conjectured that P(q, h) < c(s) q1 +‘ and in 1936 P. Turén (6) proved, assuming the grand Riemann Hypothesis, that, given 6 > 0, the number of arithmetic progressions mod q for

which P(q, h) < ¢(q)log“"q is asymptotically ¢(q) as q —-> oo. Linnik (4) proved, without appeal to any unproved conjecture, that P(q,h) < qc, where C is an absolute positive constant!“ We prove here THEOREM. Given any 6 > 0, there exists a sequence Q of zero density (which may be empty) such that the number of arithmetic progressions h mod q for

which P(q, h) < ¢(q) (log q)1 +" is asymptotically ¢(q) as q —» 00 through integers not in Q.

Proof. It will be evident from what follows that we obtain a result rather more precise than that given in the statement of the Theorem. We set ofi‘ from the theorem of Davenport—Halberstam (2), as sharpened in Gallagher (3, Theorem 3), which may be stated in the following form:

If

logp,

Z

9(x;q, h) =

and

A > 0,

,2
*lXOOg iX)”‘2’3"’ > x provided X 2 X0((5);

hence 0(x;q, h) = 0 if qX.

It follows from (1)

with A = 5/2 that

x’

(11(4)

2:2

WW1? 00q < (logxrm’ whence, if IQXI denotes the cardinality of Qx,

X(log X)""

X

'9“ < (10n \ W

(2)

To prove the theorem it suffices to take no

Q =

U aIll = l

and to show that

N'1 Z Isl—+0

as N—>oo.

ZMQN

But (2) implies that the expression on the left is < (log N)"’/", so that the proof of the Theorem is complete.

Recently H. L. Montgomery (5) has found, by different and deeper methods,

an improvement of the results cited from (2) and (3), which allows us to assert, in place of (l), the following:

x —2 < X < x, log x

‘If

2

q

then

«a

2

{(9064.11)

_x_} < Xxlogx.’

¢(q)

(3)

THE LEAST PRIME IN AN ARITHMETIC PROGRESSION

61

With the help of this result we can improve our Theorem. Letf (y) be a positive monotonic increasing function Of y, such that f (y) < logy and f (y) tends arbitrarily slowly to infinity as y —> 00. If we take x = X f*(X) log X in

(3) and argue in essentially the same way as we did above, we find even that

P(q, h) < q f(q)10gq for asymptotically ¢(q) progressions h mod q, for almost all q. References

awaype

. Q-IOWLA, S. J. Indian Math. Soc. (2), 1 (1934), 1—3. DAVENPORT, H. and HALBERSTAM, H. Michigan Math. J. 13 (1966), 485—489. GALLAGHER, P. Mathematika 14 (1967), 21—27.

. LINNIK, JU. V. Mat. Sbornik 15 (1947), 139—178. MONTGOMERY, H. L. Michigan Math. J. 17 (1970), 33—39. . TURAN, P. Acta Sci. Math. Szeged 8 (1937), 226—235.

Polarized Partition Relations for Ordinal Numbers P. Ekpos, A. HAINAL, AND E. C. MILNER* 1. Introduction

There have been several instances where some particularly well-chosen symbol has enhanced the development of a branch of mathematics, and the

partition symbol at —) (a0, a1)’

(1.1)

invented by Richard Rado is a case in point. By definition, (1.1) means that the following relation between the ordinal (or cardinal) numbers a, a0, a, holds: If A is an ordered set of order type a (we shall write tp A = 0:) and if [A]’ = {X r: A : [XI = r} is partitioned in any way into two sets K0 and K1,

then there are p < 2 and A’ c A such that tp A’ = at, and [A']" c KP. Erdo‘s and Rado were the first to realize that a large number of seemingly unrelated problems in set theory could be reduced to a question of deciding whether or

not some partition relation like (1.1) holds. In (5) and (6) they began a systematic study of these relations and laid the foundations of what they called a partition calculus to serve as a kind of unifying principle in set theory. Since these two pioneer papers several others have been written on the subject. In

particular we refer to the long paper by Erdos, Hajnal, and Rado (3) which contains an almost complete analysis for partition relations involving infinite cardinal numbers. Rado’s compact symbol (1.1), which reveals at a glance the

whole content of a fairly complicated combinatorial statement, proved to be particularly convenient and flexible for the development of this calculus. Apart from the merit of compactness, the symbol enjoys other advantages. The negation of any statement (1.1) is conveniently expressed by replacing the arrow —> by a non-arrow 4—». The symbol has the following obvious monotonicity properties, if a’ 2 a, ’3’ S [3 and y’ < y, then (1.1) implies that a! _) (fil, ,yl)l'.

The arrow in (1 . 1) separates the two kinds of monotonicity involved and this is * Research supported by NRC grant A—5198. Part of this paper was written at the Vancouver branch of the Canadian Mathematical Congress 1969 Summer Research Institute.

63

64

P. names, A. HAJNAL, AND E. c. MILNER

helpful in recognizing which relations are best possible. Finally, the symbol readily lends itself to a number of interesting generalizations (see (6) and (3)).

In this paper we investigate one of these generalizations, the so-called polarized partition symbol. We consider only the simplest of such relations, namely those of the form

a

{/3}

—>

a0 a1} 1-1

.

{Bo [31

(1.2)

By definition, this means that: IfA and B are ordered sets, tp A = oz, tp B = [3,

and if the cartesian product A x B is partitioned in any way into two sets Ko and K1, then there are p < 2 and sets A’ c A, B’ c B such that tp A’ = at”, tp B’ = fl, and A’ x B’ c Kp. Ifone considersinstead, partitions of [A]' x [B]’ for arbitrary integers r, s, the corresponding relation is represented by replacing the exponents 1,1 in (1.2) by r, s; these more general polarized relations clearly include the ordinary partition relations (1.1). Since we only consider relations with the exponents 1,1, for the remainder of this paper we shall omit these from (1.2) and simply write

{ [3“l I“30 31“‘1 —*

.

Note that, as for the ordinary partition symbol, the negation of (1.2) is expressed by replacing —> by 4—». Also, we have the same monotonicity properties: ifot’ 2 a, fi’ 2 [3, ap’ S up, fip’ S [3,, (p < 2), then (1.2) implies that

{21%12122111 Polarized partition relations were first introduced in (6), and in (3) a number

of these relations involving cardinal numbers were established. As we already remarked, the theory for the ordinary partition relations involving cardinal

numbers is fairly complete, but for polarized relations the situation is very different. There remain unsolved problems involving only the smallest trans-

finite cardinal numbers. For example, it is not known if the relation

{Ni} N2

—-y

{N1 N1: No No

is true or false. In this paper we shall establish relations of the form (1.2) which involve ordinal numbers. As a starting point for our investigation we mention the simple, but slightly

surprising, negative relation 1 H (wnm)ln

2 wla)

cu } (”1&2

true or false?

There is another problem of this kind (see §4) which we cannot settle. 1' Where we use the continuum hypothesis to prove a result, we prefix the statement by (*) for easier recognition.

POLARIZED PARTITION RELATIONS FOR ORDINAL NUMBERS

67

PROBLEM 2. Does the relation

1"“ :~ 3 1: mlo)+2

:w1m+1€

hold for a < a), andé < calm”?

It follows as a special case of a result proved in (2, Theorem 1) that col to1 {mi} _. {a1 any}

(1.13)

if a < w 1, y < co2 and co(co,”) = (0,. We also showed (2, Theorem 2) that m1 {(017} ++ {w 1H a), «21’ i

(1.14)

if co(col7 = a). The method we used to prove (1.13) is very different from the methods used in this paper. We shall not give the details, but with the same method used in (2) one can also show that co a) {(09} _, {ca1 (9.17}

(y < (0,).

(1.15)

We mention these results because these three relations (1.13)—(1.15), together with (1.4), (1.5), (1.7), (1.8), (1.9) and (1.10) give a complete analysis of the symbol (1.2) for the case a = (0,, B = (01" and a0 = 1. In §6 we establish some strong negative results. Using the continuum hypo-

thesis we prove (Theorem 7) that, for y < (02 and c0601") = (0,, co1

(*)

l

2

to,’

wln’“ + 1

a)

col”

a)

.

,

v

v

4—»

1

(1.16)

(01’

Here we are using the partition symbol with alternatives (for the definition see §2). An equivalent formulation of (1.16) is the following. If tp S = (017 < (02 and the continuum hypothesis is assumed, then there is a family of N1 sets

Fn c S(,u p, forall neNou

UM}.

Then M’,, is a cofinal subset of W1(k 6N1). Since N, is infinite, it follows from

the induction hypothesis that there are elements [1,, e M’,,(n e N,) such that

tp {flu mam} 2 This defines [1,, e M, for all n < (0. By the construction,

tp{p,,:n < w}=12tp{p,,:neN,}> Z a, = to”.