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COMBINATORICS, Volume http://dx.doi.org/10.1090/pspum/019 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XIX COMBINATORICS AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 1971 Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society Held at the University of California Los Angeles, California March 21-22, 1968 Prepared by the American Mathematical Society under National Science Foundation Grant GP-8436 Edited by Theodore S. Motzkin AMS 1970 Subject Classifications Primary 05Axx, 05Bxx, 05Cxx, 10-XX, 15-XX, 50-XX Secondary 04A20, 05A05, 05A17, 05A20, 05B05, 05B15, 05B20, 05B25, 05B30, 05C15, 05C99, 06A05, 10A45, 10C05, 14-XX, 20Bxx, 20Fxx, 50A20, 55C05, 55J05, 94A20 International Standard Book Number 0-8218-1419-2 Library of Congress Catalog Number 74-153879 Copyright © 1971 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government May not be produced in any form without permission of the publishers Leo Moser (1921-1970) was active and productive in various aspects of combin• atorics and of its applications to number theory. He was in close contact with those with whom he had common interests: we will remember his sparkling wit, the universality of his anecdotes, and his stimulating presence. This volume, much of whose content he had enjoyed and appreciated, and which contains the re• construction of a contribution by him, is dedicated to his memory. CONTENTS Preface vii Modular Forms on Noncongruence Subgroups BY A. O. L. ATKIN AND H. P. F. SWINNERTON-DYER 1 Selfconjugate Tetrahedra with Respect to the Hermitian Variety xl+xl + *l + ;cg = 0 in PG(3, 22) and a Representation of PG(3, 3) BY R. C. BOSE 27 Multipartitions and Multipermutations BY M. S. CHEEMA AND T. S. MOTZKIN 39 Simplicial Geometries BY HENRY H. CRAPO AND GIAN-CARLO ROTA 71 Problems and Results in Combinatorial Analysis BY P. ERD5S 77 Multirowed Partitions with Strict Decrease along Columns (Notes on Plane Partitions. IV) BY BASIL GORDON 91 Rota's Geometric Analogue to Ramsey's Theorem BY R. L. GRAHAM AND B. ROTHSCHILD 101 Combinatorial Representations of Abelian Groups BY ALFRED W. HALES 105 Designs with Transitive Automorphism Groups BY MARSHALL HALL, JR 109 Truncated Finite Planes BY HAIM HANANI 115 Homogeneous 0-1 Matrices BY ALEXANDER HURWITZ 121 The Greedy Algorithm for Finitary and Cofinitary Matroids BY VICTOR KLEE 137 Collections of Subsets Containing no Two Sets and Their Union BY DANIEL KLEITMAN 153 A Combinatorial Method for Embedding a Group in a Semigroup BY N. S. MENDELSOHN 157 Asymptotics of Tournament Scores BY LEO MOSER 165 Sorting Numbers for Cylinders and Other Classification Numbers BY THEODORE S. MOTZKIN 167 v vi CONTENTS Pathological Latin Squares BY E. T. PARKER 177 Some Problems in the Partition Calculus BY RICHARD RADO 183 Solution of Kirkman's Schoolgirl Problem BY D. K. RAY-CHAUDHURI AND RICHARD M. WILSON 187 A Generalization of Ramsey's Theorem BY BRUCE ROTHSCHILD 205 Nonaveraging Sets BY E. G. STRAUS 215 On (k, /)-Coverings and Disjoint Systems BY J. D, SWIFT 223 (1, 2, 4, 8)—Sums of Squares and Hadamard Matrices BY OLGA TAUSSKY 229 Dichromatic Sums for Rooted Planar Maps BY W. T. TUTTE 235 Author Index 247 Subject Index 251 Preface Combinatorics is the theory of finite sets. This is a wide, amorphous, primordial subject matter which in principle includes subareas where more structure and more specific structure is assumed; but whether historically they have grown before or next to combinatorics, or were part of a more general theory encompassing similar infinite structures, or were indeed originally part of combinatorics, many such areas are now considered as separate fields. Number theory and the theories of partitions and finite groups are examples of the first; the theories of finite fields and finite geometries of the second; graph theory and the theory of designs are on their way to be examples of the third kind. Because of the recent symposia on graph theory, and the existing collections on applied combinatorics (combinatorial geometry, probability theory and extra- mathematical applications), it was decided to emphasize at the present symposium the theory of simple general or homogeneous structures. Of the twenty-four talks, eight treat general structures, nine treat designs (homogeneous structures), six treat applications of the first two topics to sets of integers, algebra and complex analysis, and one is a survey article mainly on general structures and partly on sets of integers. (Asymptotic results occur in seven of the thirteen papers on general structures or applications thereof; computers were used in three papers.) Thus the scope (structures, systems, applications) is close to that of Series A (as of 1971) of the Journal of Combinatorial Theory (Series B will be on graph theory). Specifically, the first class (general structures) includes the papers of Kleitman, Motzkin, Swift on families (sets of sets), of Crapo/Rota and Klee on simplicial geometries and pregeometries (matroids) or pregeometries, of Tutte on planar graphs, Rothschild on Ramsey theorems for graphs, and Rado on transfinite Ramsey theorems. The second class (homogeneous structures) includes the articles of Hurwitz on 0-1 matrices, Parker on latin squares, Ray-Chaudhuri/Wilson on Kirkman designs, Hall on designs and groups, Mendelsohn on graphs, semigroups, and groups, Hales on trees and Abelian groups, Bose and Hanani on designs related to finite geometries, and Graham/Rothschild on Ramsey theorems in finite geometry. Investigations close to the first class are found in the papers of Gordon and Cheema/Motzkin on partitions, Moser and Straus on sets of integers; to the second class in the articles of Taussky on algebraic identities connected with Hadamard matrices, and Atkin/Swinnerton-Dyer on modular forms for discrete groups. Erdos' survey article deals in sections I, III, IV with families, and in sections II, V, VI with applications including geometry and sets of integers. I hope that this volume, with its numerous and varied open questions and new vii viii PREFACE methods and results, extending from the solution of a century-old problem on designs to algebro-geometric and number and function theoretic studies, will adequately reflect work done and in progress and contribute to growth and change in combinatorics. Acknowledgments On behalf of the contributors and participants, the editor wishes to express gratitude and appreciation to the American Mathematical Society for constant and multifaceted cooperation, to the National Science Foundation for financial support, to the University of California, Los Angeles, for the use of its facilities and to William Clowes & Sons, Ltd. for the excellent transformation into book form. Theodore S. Motzkin AUTHOR INDEX Roman numbers refer to pages on which a reference is made to an author or a work of an author. Italic numbers refer to pages on which a complete reference to a work by an author is given. Boldface numbers indicate the first page of the articles in the book. Abramowitz, M., 172,173 Crapo, H. H., 71, 72, 75,137,138,151 Adams, J. F., 231, 232,233 Crawley, P., 106,108 Ahlfors, L. V., 116,120 Czipszer, J., 86,87 Ahrens, W., 188,203 Aleksandrov, P. S., 71, 75 Danzer, L., 80,87 Anstice, R. R., 188,202 Davis, E.W., 188,203 Asche, D. S., 143,151 Daykin, D. E., 85,86 Ashworth, M. H., 2,25 Debrunner, H., 80,88 Ativan, M. F., 233 Dickson, L. E., 118,119,120 Atkin, A. O. L., 1, 21,25 Dieudonne, J., 233 Auluck, F. C, 39, 69,100,100 Dilworth, R. P., 78,87 Dixon, A. C, 188,203 Baines, M. J., 85,86 Dlab, V., 137,148,151,151 Ball, W. W. R., 188, 189, 193, 196, 201, 203 Dudeney, H. E., 188,203 Behrend, F., 77,86 Dushnik, B., 184,185 Bill, S., 188,203 Eckenstein, C, 188,202,203 Bleicher, M. N., 137,151,151 Edmonds, J., 137,143,146,151 van derBlij,F., 233 Eichhorn, W., 229,231,233 Boole, G., 167,176 Elliott, P. D. T. A., 80,87 Bose, R. C, 27, 27, 28, 29, 30, 37, 109, 112, Erdos, P., 51, 69, 77, 77, 78, 79, 80, 81, 82, 113,115,120, 134, 135, 188, 189, 190, 195, 84, 86, 86, 87, 88, 153, 155, 183, 184, 185, 196,201 216, 220,222 Bott, R., 229,233 Euler, L., 39, 69, 111, 180,181 Bray, A., 188,203 Fine, N. J., 50, 69 Brooks, R. L., 237,245 Finlayson, H. C, 167,176 Brualdi, R. A., 137,151 Folkman, J., 233 Bruck, R. H., 112,113, 111, 181 Fort, M. K., Jr., 223, 225,228 deBruijn,N.G., 80,87 Fricke, R., 3, 6 Buck, R. C, 161,164 Frost, A., 188,203 Burnside,W., 188,203 Fueter, R., 231,233 Fujiwara, M., 176,176 Carlitz, L., 40, 50, 69 Fulkerson, D. R., 146,151 Carmichael, R. D., 116,120 Carpmael, E., 188,203 Gale, D., 137,151 Catalan, E., 167,176 Gerstenhaber, M., 233 Cayley, A., 188,202 Glaisher, J. W. L., 39,53,69 Chakravarti, I. M., 27, 28, 30,37 Gleason, A. M., 81,88 Chaundy,T., 69 Gordon, B., 40,50, 69,91,91,96,98,99, 100 Cheema, M. S., 39,40, 69, 69 Graham, R. L., 101 Clatworthy, W. H., 29,37 Griinbaum, B., 80,87 Connor, W.S., Jr., 134,135 Graver, J. E., 81,88 247 248 AUTHOR INDEX Greenwood, R. E., 81,88 Marsden, E., 188, 203 Griinbaum, B„ 80,87 Meinardus,G., 39,69 Gupta, H., 69 Mendelsohn, N.S., 157 Gwyther, A. E., 69 Mertelsmann, A. F. H., 188, 203 Miller, E. W., 82,88,184,185 Hadwiger,H.,80,88 Miller, J.
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