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. C. P., 69 Hajnal, J., 81, 82, 86,87,183,184,185 Millington, M. H. (see Ashworth, M. H.) Hales, A. W.,88,104,104,105,106,108 Milnor, J., 229, 233 Hall, M., Jr., 109, 113, 115, 116, 117, 120, Minty,D.J., 144,151 121,134,135, 174, 176, 111, 180, 181, 188, Moon, J., 165,166,166 189,195,196,202 Moore, E. H., 188, 203 Hanani, H., 80, 83, 88, 115, 115, 116, 120, Moser,L., 165,172,173,176 121,135,188,189,192, 202, 223, 228 Moser, W. O. J., 80, 84,88 Hansen, S., 88 Motzkin, T. S., 39, 40, 69, 69, 80, 88, 167, Harary, F., 107,108,108 176,205,272 Hardy,G.H„39,69 Mullin, R. C, 235, 245 Harper, L. H., 173,176 Hedlund, G. A., 223, 225,228 Nanda, V. S., 39, 58, 70 Higgs, D. A., 137,152 Nash-Williams, C. St. J. A., 88 Higman, D. G., 109, 111, 112,113 Neumann, B. H„ 161,164 Houten, J., 91,96,98, 99,100 Newman, M. H. A., 233 Hughes, N. J. S., 137,151,151 Ore, O., 212, 213 Hurwitz, A., 121,121,126,132,135, 229,233 Paige, L. J., 180,181 Igusi, Jun-ichi, 21,25 Pall, G., 232, 233 Jngham, A. £., 58, 69 Parker, E. T., 115,120, 177,177,181, 202 Peirce, B., 188, 202 Jewett,R. I., 88,104,104 Pfister, A., 229, 233 John, P. W. M., 134,135 Phillips, R. S., 231, 232, 233 Power, J., 188, 203 Katona, Gy., 79,88 Preston, G. B., 137,151 Kelly, ^.M., 80,88 Prins, G., 107,108,108 Kertesz, A., 137,151 Putter, J., 233 Kirkman, T. P., 188,202,203 Klee,V,L., 80,88,137 Rado, R., 78, 81, 87, 88, 137, 143, 149, 151, Klein, F., 3,6,25 151,153,155, 183,183,184,185 Kleitman, D., 77, 79, 88, 153, 153, 155, 166 Ramanujan, S., 39, 69 Ko,Chao, 78,87,153,155 Ramsey, F. P., 80, 81, 88,183,185, 213 Kruskal, J. B., Jr., 137,151,151 Rao, C.R., 189,202 Rao, V. R., 134,135 Lane, H , 109,113 Ray-Chaudhuri, D. K., 187 Lax, P. D., 231, 232,233 Reiss, M., 83,88 Lea, W-, 188,203 Rieger,G.J.,39,53, 70 Leech, J., 16,25 Robertson, A. P., 137, 143,152 Lehmer, p. H., 68,69 Robertson, M. M., 39, 58, 70 Lehner,J.,21,25,51,69 Rogers, L. J., 97,100 van tint, J. H., 84,88 Rota, G.-C., 71, 71, 72, 75, 137, 138, 151, 167,176 MacMahon, P. A., 40,45, 69,91, 95,100 Roth, K. F., 84, 88 Mann, H. B„ 177,179,180,181,202 Rothschild, B., 101,104, 205, 213 Marczewski, E., 137,151,151 Ryser, H. J., 131,135, 111, 181,202,233 AUTHOR INDEX 249 Sarkozi, A., 84, 89 Taussky, O., 229, 229, 230, 232, 233 Schmidt, J., 137,152 Todd, J. A., 70 Schmidt, W., 82, 83, 89 Tompkins, C. B., 68, 70 Schonheim, J., 223, 225, 228 Tutte, W. T., 235, 235, 237, 245 Schur, I., 85, 89 Scrimger, E. B., 137,151 van der Waerden, B. L., 83, 89 Serre, J.-P., 21, 25 Wales, D., 109,113 Shimamoto, T., 28, 37,109,112,113 Watson, G.N., 245, 245 Shrikhande, S., 115, 120, 188, 189, 190, 201 Weil, A., 24, 25 Sloane, N. J. A., 172,173,176 Wells, M. B., 68, 70 Smith, C. A. B., 237,245 Welsh, D. J., 137,152 Spencer, J., 231,233 Weston, J. D., 137,143,149,152 Sperner, A., 77, 89,154,155 Weyl, H., 100 Stegun, LA., 172,173 Whitney, H., 72, 75, 143,148,152 Steiner, J., 188, 202 Wielandt,H., Ill, 113 Stiefel,E.,231,233 Wilson, R. M., 187 Stone, A. H., 237, 245 Wohlfahrt,K., 2, 25 Straus, E. G., 215, 216, 220, 222 Woolhouse, W. S. B., 188, 202, 203 Swift, J. D., 223 Wright, E. M., 39, 50, 53, 56, 65, 70 Swinnerton-Dyer, H. P. F., 1 Wyman, M., 172, 173, 176 Sylvester, J. J., 39, 53, 70,188, 202, 203 Yackel,J.,81,88 Szekeres, G., 40, 70, 79, 80, 81,88,89 Szemeredi, E., 84, 89 Zassenhaus, H., 229, 233 SUBJECT INDEX abelian group, 105 Cauchy-Riemann equations, 230 adjoinable, 134 Cayley numbers, 229 affine geometry, finite, 117 chain, 205 affine space, 101 ascending, 205 Alexander duality, 75 characteristic numbers, 40 algebra, Clifford, 231 characteristic polynomial, 174 composition, 229 characteristic value problems, 232 algebraic differential equations, 176 circuit, 137,138 algebraic function, 4 circuit clutter, 146 algebraic geometry, 25 class, similarity, 107 algebraic variety, 72 classification anticommuting, 229 1-level, 169 arithmetic progression, 83, 215, 220 2-level, 169 ascending chain, 205 3-level, 169 associate classes, 111 classification number, 167 association schemes, 28 Clifford algebra, 231 asymptotic expressions, 173 closure operator, 73 asymptotic behavior, 52 clutter, 142 asymptotic theory, 39 circuit, 146 asymptotics, 165 complementary, 142 attachment, point of, 157 cofinitary matroid, 137 vertex of, 157 cofinite, 138 coloring, 74 balanced incomplete block design, 115, 121, column-equivalent, 133 187,188 combinatorial geometry, 72 base block methods, 195 combinatorially distinct, 235 bases, 137,138 compactness, 205 equipollence of, 137,149 complete linear graphs, 74 basis, 75 complete set, 177 Bell-Stirling numbers, 167 completed resolvable design, 189 Bernoulli numbers, 171 completeness, 227 Bijections, 168 composition, 169 bipartition numbers, 39 composition algebra, 229 Block design, 109 composition for sums of squares, 229 balanced incomplete, 115,121,187,188 computable, 178 resolvable, 187,188 computer, 1, 83 resolvable sub-, 188 computer program, 12 symmetric, 121,188 computing, 158 partially balanced, 109 configuration, Desargues, 73 symmetric, 109 congruence properties, 22,173 block, 83 continued fraction, 99 blocked, 205 Conway diagram, 15 Brown, John W., 177 covering, (k,l)-, 223 Bruck, R. H., 177 covering system, 223 covers, 167 C-design, 115 crystal surfaces, 100 cardinal numbers, 183 cusp form, 3
251 252 SUBJECT INDEX cyclic composition, 169 exponential generating function, 171 cyclic group, 180 extremal problems, 86 cyclic latin square, 177 cyclic partition, 169 false ^-function, 91 cyclic permutation, 166 finitary matroids, 137 cycloidal, 10 finite afline geometry, 117 A-system, 82 finite basis property, 72 depend,138 finite graded graph, 206 dependent, 138 finite inversive geometry, 117 Desargues configuration, 73 finite projective plane, 121,126,133,177, design, C-, 115 188 completed resolvable, 189 first order differential equations, 171 Kirkman, 187 Fisher's inequality, 131 orbital, 109 flat, 72 pairwise balanced, 188 Frobenius' theorem, 229 diagram Fueter equation, 231 Conway, 15 function (see also generating function) subset, 206 function, algebraic, 4 subspace, 207 enlarging, 138 dichromatic polynomial, 236 false 0-, 91 dichromatic sums, 235 isotonic, 138 difference methods, 195 Whitney rank, 72 difference set, 112 functional equation, 107, 237 differential equations, algebraic, 176 fundamental domain, 3 differentials, 23 Dirac equation, 231 Galois field,11 6 directed graph, 157 Gauss's identity, 99 directed path, 206 general continuum hypothesis, 183 divisors of zero, 229 generalized Steiner problem, 223 dual, 138 generating functions, 41, 237 dual of an operator, 138 exponential, 171 duality, 75, 237 umbral, 167 Alexander, 75 geometric lattice, 72,101 elliptic curve, 24 geometry, 71 embedding, 74,157 algebraic, 25 entire, 3 combinatorial, 72 equation finite, 117 Cauchy-Riemann, 230 affine, 117 functional, 107, 237 inversive, 117 differential simplicial, 75 algebraic, 176 "graded" graph, 206 first order, 171 graph, 105, 206 Laplace, 230 complete linear, 74 Dirac, 231 directed, 157 Fueter, 231 finite graded, 206 equipollence of bases, 137,149 inclusion, 123,124,126,127,128,134 equivalent, 122 J-inclusion, 124,130 ErdoV conjecture, 209 locally finite,20 6 Euler identity, 44 greedy algorithm, 137 Euler-Poincare formula, 71 group, 109,157,180 exchange property, basis, 72 abelian, 105 Steinitz-Mac Lane, 72 cyclic, 180 SUBJECT INDEX 253 Hadamard matrix, 229 fists, 167 Hall, Marshall, 180 loop-map, 236 Hecke operators, 1 map Hermitian curve, 27 link-, 236 Hermitian matrix, 27 loop-, 236 Hermitian variety, 27 rooted planar, 235 histogram, 165 vertex, 236 homeomorphism, 235 matrix homogeneous Hadamard, 229 ;-, 122 Hermitian, 27 ;, *-, 122 incidence, 106,121,126,133 homology, 71 permanent doubly stochastic, 83 hyperplane, 138 matroid, 137,141 cofinitary, 137 identities in semigroups, 170 finitary, 137 incidence matrix, 106,121,126,133 maximal, 123,126,129,131,132,133 inclusion graph, 123,124, 126, 127, 128, 134 J-, 124 indecomposable edge solutions, 134 maximality, 134 indecomposable solutions, 134 merging independent, 138 symmetric, 168 infinite almost directed path, 208 subgroupwise, 168 infinite directed path, 206 methods initial ordinal, 184 base block, 195 injection, 168 difference, 195 in valence, 157,168 minor, 180 modular form, 1 J-equivalent, 123 modular group, 1 ./-homogeneous, 122 modular periodicity, 176 ./-inclusion graph, 124,130 Monte Carlo, 9 j, &-homogeneous, 122 Motzkin's theorem, 206 J-maximal, 124,130 multicompositions, 40 Jacobi identity, 40 multipartition, 39 juxtaposition, 121,131,132,133 multipermutations, 40 multirowed partitions, 91 Kirkman design, 187 Kirkman's schoolgirl theorem, 187 iV-ads, 168 (k,l) -covering, 223 iV-tuples, 168 (k, i) -disjoint system, 223 nonaveraging sets, 215 Konig's theorem, 212 noncongruence subgroups, 1 nondividing sets, 215 Laplace differential equation, 230 nullity, 75 latin square, 177 numbers, 167 cyclic, 177 Bell-Stirling, 167 pathological, 177 Bernoulli, 171 transversal of a, 178 bipartition, 39 latin square of order ten, 177 cardinal, 183 lattice, geometric, 72,101 Cayley, 229 lattice of subspaces, 207 characteristic, 40 Lie group, p-adic, 24 classification, 167 linear inequalities, 72 ordinal, 184 link-map, 236 partition, 167 listed partition, 169 Ramsey's, 81 listed sorting, 169 theory, 83 254 SUBJECT INDEX obstructed, 205 pregeometry, 72,138 1-level classification, 169 prime decomposition, 174 operator, 137 prime field, 174 dual of an, 138 probabilistic, 84 orbital design, 109 projective plane, finite,121,126,133 , order types, 184 177,188 ordinal, initial, 184 projective space, 101 ordinal number, 184 proper, 169 orthogonal, 177 property B, 82 orthogonal array, 189 resolvable, 190 quaternions, 229 orthogonal rank, 75 orthogonality, 72 Ramsey's numbers, 81 out-valence, 157 Ramsey's theorem, 81,101,183, 205 reciprocity theorem, 93 p-adic Lie group, 24 recurrence, 170 packing, 167 relations, 40, 45 Paige, L. J., 180 recursive calculation, 237 pairwise balanced design, 188 regular, 3 parallel class, 188 representation, 27,105 parameters, 188 resolvable balanced incomplete block partially balanced block design, 109 designs, 187 partition, 74,169 resolvable orthogonal array, 190 cyclic, 169 root, 235 listed, 169 root-edge, 235 multi-, 39 root-face, 235 multirowed, 91 root-vertex, 235 plane, 40,91 rooted planar map, 235 set, 167 rooted tree, 105 solid, 40 Ryser, H. J., 177 partition calculus, 183 partition function, 107 partition number, 167 Schonheim