Algebraic, Extremal and Metric Combinatorics 1986

257 Pages·1988·9.151 MB·English

by  M. M. Deza, P. Frankl, I. G. Rosenberg

LONDON MATIIEMATICAL SOCIETY LECfURE N01E SERIES
Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge, 16 Mill Lane, Cambridge CB2 ISB, England
The books in the series listed below are available from booksellers, or, in case of difficulty,
from Cambridge University Press.
4 Algebraic topology, J.F. ADAMS
17 Differential germs and catastrophes, Th. BROCKER & L. LANDER
27 Skew field constructions, P.M. COHN
34 Representation theory of Ue groups, M.F. AT IYA H et al
36 Homological group theory, C.T.C. WALL(ed)
39 Affine sets and affine groups, D.G. NORTIICOTT
40 Introduction to Hp spaces, P J. KOOSIS
42 Topics in the theory of group presentations, DL. JOHNSON
43 Graphs, codes and designs, P J. CAMERON & J.H. VAN LINT
45 Recursion theory: its generalisations and applications, F.R. DRAKE & S.S. WAINER (eds)
46 p-adic analysis: a short course on recent work, N. KOBLITZ
49 Finite geometries and designs, P. CAMERON, J .W. P. IDRSCHFELD &
D.R. HUGHES (eds)
SO Commutator calculus and groups of homotopy classes, HJ. BAUES
51 Synthetic differential geometry, A. KOCK
54 Markov processes and related problems of analysis, E.B. DYNKIN
51 Techniques of geometric topology, R.A. FENN
58 Singularities of smooth functions and maps, J.A. MARTINET
59 Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI
60 Integrable systems, S.P. NOVIKOV et al
62 Economics for mathematicians, J.W.S. CASSELS
65 Several complex variables and complex manifolds I, MJ. FIELD
66 Several complex variables and complex manifolds n, MJ. FIELD
68 Complex algebraic surfaces, A. BEAUVILLE
69 Representation theory, I.M. GELFAND et al
74 Symmettic designs: an algebraic approach, E.S. LANDER
76 Spectral theory of linear differential operators and comparison algebras, H.O. CORDES
11 Isolated singular points on complete intersections, EJN. LOOIJENGA
78 A primer on Riemann surfaces, A:F. BEARDON
79 Probability, statistics and analysis, J:F.C. KINGMAN & G.E.H. REUTER {eds)
80 Introduction to the representation theory of compact and locally compact groups,
A. ROBERT
81 Skew fields, P K. DRAXL
82 Surveys in combinatorics, E.K. LLOYD {ed)
83 Homogeneous structures on Riemannian manifolds, F. TRICERRl & L. VANHECKE
84 Finite group algebras and their modules, P. LANDROCK
85 Solitons, P.G. DRAZIN
86 Topological topics, I.M. JAMES (ed)
87 Surveys in set theory, A.R.D. MATHIAS {ed)
88 FPF ring theory, C. FAITH & S. PAGE
89 An F-space sampler, NJ. KALTON, N.T. PECK & J.W. ROBERTS
90 Polytopes and symmetry, S.A. ROBERTSON
91 Classgroups of group rings, MJ. TAYLOR
92 Representation of rings over skew fields, A.H. SCHOFIELD
93 Aspects of topology, I.M. JAMES & E.H. KRONHEIMER {eds)
94 Representations of general linear groups, G.D. JAMES
95 Low-dimensional topology 1982, R.A. FENN (ed)
96 Diophantine equations over function fields, R.C. MASON
97 Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN
98 Localization in Noetherian rings, A. V. JATEGAONKAR
99 Methods of differential geometry in algebraic topOlogy, M. KAROUBI & C. LERUSTE
100 Stopping time techniques for analysts and probabilists, L. EGGH E
101 Groups and geometry, ROGER C. LYNDON
103 Surveys in combinatorics 1985, I. ANDERSON (ed)
104 Elliptic sttuctures on 3-manifolds, C.B. THOMAS
105 A local spectral theory for closed operatorS, I. ERDELY I & WANG SHENGWA NG
106 Syzygies, E.G. EVANS & P. GRIFFITH
107 Compactification of Siegel moduli schemes, C-L. CHAI
108 Some topics in graph theory, H.P. YAP
109 Diophantine Analysis, J. LOXTON & A. VAN DER POORT EN (eds)
110 An introduction to surreal numbers, H. GONSHOR
111 Analytical and geometric aspects of hyperbolic space, D.B.A.EPSTEIN (ed)
112 Low-dimensional topOlogy and Kleinian groups, D.B.A. EPSTEIN (ed)
113 Lectures on the asymptotic theory of ideals, D. REES
114 Lectures on Bochner-Riesz means, K.M. DAVIS & Y -C. CHANG
115 An introduction to independence for analysts, H.G. DALES & W.H. WOODIN
116 Representations of algebras, P.J. WEBB (ed)
117 Homotopy theory, E. REES & JD.S. JONES (eds)
118 Skew linear groups, M. SHIRVA NI & B. WEHRFRITZ
119 Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL
120 Lectures on Fermat varieties, T. SillODA
121 Proceedings of Groups-St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds)
122 Non-classical continuum mechanics, RJ. KNOPS & A.A. LACEY (eds)
123 Surveys in combinatorics 1987, C. wmTEHEAD (ed)
124 Lie groupoids and Lie algebroids in differential geometry, K. MACKENZm
125 Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZm
126 Vander Corput's method for exponential sums, S.W. GRAHAM & G. KOLESNIK
127 New directions in dynamical systems, T J. BEDFORD & J.W. SWIFT (eds)
128 Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU
129 The subgroup sttucture of the fmite classical groups, P.B. KLEIDMAN & M.W LffiBECK
130 Model theory and modules, M. PREST
131 Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds)
132 Whitehead groups of fmite groups, ROBERT OLIVER
London Mathematical Society Lecture Note Series. 131
Algebraic, Extremal and Metric
Combinatorics, 1986
Edited by
M-M. Deza & P. Frankl, CNRS, Paris
and
I.G Rosenberg, University of Montreal
The right of tlte
Uniwr-sity of Combridge
to print and ull
all manner of books
was granted by
HMry Y/11 in 1134.
T1tr Uni~"IIY has print~d
ond publislrtd Mnti111«HUiy
since U84.
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CONTENTS
Introduction vii
List of Talks viii
Participants ix
Some Recent Combinatorial Applications of Borsuk-Type Theorems 1
N.ALON
On Extremal Finite Sets in the Sphere and Other Metric Spaces 13
E. BANNAI
Metric and Geometric Properties of Sets of Permutations 39
P.J. CAMERON
Infinite Geometric Groups and Sets 54
P.J. CAMERON, M-M. DEZA & N.M. SINGH
Intersection and Containment Problems without Size Restrictions 62
P.FRANKL
Distance-Transitive Graphs of Valency k, 8 <~ k < 13 112
A.A. IVANOV & A.V. IVANOV
Latin Square Determinants 146
K.W . JOHNSON
A Computer Search for a Projective Plane of Order 10 155
C.W.H. LAM, L.H. THIEL & S. SWIERCZ
Matroids, Algebraic and Non Algebraic 166
B. LINDSTR6M
Algebraic Properties of a General Convolution 17 5
!.G. ROSENBERG
Quasi Groups, Association Schemes, and Laplace Operators on 205
Almost Periodic Functions
J.D.H. SMITH
Geometric Methods in Group Theory 219
S.D. SMITH
Problem Section 238

vii
Introduction
This book contains the invited papers of the conference Algebraic
II
combinatorics and extremal problems 11 held at the Universite de Montreal from July
28 to August 2, 1986. This was the first time a conference focusing on these two
subjects was held. The main reason for organizing such a meeting was that these
J
apparently mutually distant parts of combinatorics are becoming more and more
intertwined. This may be seen, for example, in the paper of P.J. Cameron bringing
together -among other things - the representation theory of symmetric groups and
the extremal problems of finite structures.
The conference proved quite useful in bringing together some of the best
specialists in those two fast developing areas of combinatorics as well as others from
related fields. All papers in the proceedings contain new results and most of them
provide up-to-date surveys of the respective subfields accessible to non-specialists.
There is a list of problems presented at the conference.
The organizers would like to use this opportunity to thank NSERC Canada
for the fmancial support (provided by a conference grant) and the Universite de
Montreal and its Departement de mathematiques et de statistique for hospitality.
Finally we would like to thank Antoine Deza, Lucien Haddad and Marc Rosenberg
for organizational help.
M. Deza, P. Frankl, I.G. Rosenberg
List of talks
Cameron -Metric and geometric properties of permutations
S.D. Smith -Geometric methodes in fmite group theory
Fuji-Hara -On balanced complementation of regular t-wise balanced designs
Gereb -A common proof for Kruskal-Katona type theorems
TIChy -Some combinatorial discrepancy problems
Anstee -Forbidden configurations
F.rankl -Erdos-Ko-Rado type theorems
F'Uredi -Families of fmite sets with forbidden configurations
Kalai -Algebraic shifting methods
Ceccherini -A new class of Planar Pi-spaces and some related topics
Shen -A consttuction of resolvable group divisible designs of block size 3
Colbourn -Leaves and neighbourhoods in triple systems
Rosa -Extending the concept of, compatibility for triple systems
Bannai -On extremal sets in spheres and other metric spaces
Ito -Designs in (P and Q) -polynomial associations schemes
Wilson -Decomposition of complete graphs
Alon -Some recent combinatorial applications of Borsuk type theorems
Terwilliger -A noncommutative algebra for association schemes
Hemmeter -Cliques of quadratic forms
Labelle - Quelques aspects de la theorie des especes en combinatoire
enumerative
Johnson -Latin square determinants
Rosenberg -General convolutions motivated by designs (with M.-M. Deza)
LindstrOm -Matroids, algebraic and non-algebraic
J.D.S. Smith -Infinite groups and finite quasigroups
Wmlder -Optimal realization of distance matrices
Zhou -The chromatic difference sequence of the canesian product of graphs
Deza -Rigid pentagons in hypercubes
Lam -Search for the projective plane of Older 10
ix
P3rtjejpapts
1. N. Alon (Tel Aviv & Bell Core) 23. J. Labelle (UQAM)
2. R. Anstee (UBC) 24. C. Lam (Concordia)
3. D. Avis (McGill) 25. B. Lindstrom (Stockholm)
4. E. Bannai (Ohio State) 26. R. Mathon (Toronto)
5. A. Berman (Technion) 27. A. Rosa (McMaster)
6. I. Cahit (McMaster) 28. I. G. Rosenberg (U.de M)
7. P. Cameron (Oxford) 29. G. Sabidussi (U.de M)
8. V. Ceccherini (Rome) 30. A. Sali (Ohio State)
9. C. J. Colburn (Waterloo) 31. H. Shen (Shangai)
10. A. Deza (Neuilly) 32. J.D.H. Smith (Iowa State)
11. M:M. Deza (Paris) 33. S.D. Smith (Illinois)
12. P. Frankl (Paris) 34. C. Tabib (E. Montpetit)
13. R. Fuji-hiU"a (Tsukuba) 35. C. Tardif (U.de M)
14. Z. Fiiredi (A.T.&T. Bell Labs) 36. P. Terwilliger (Wisconsin)
15. M. Gereb (Harvard) 37. R. Tichy (Vienna)
16. A. Guelzow (Manitoba) 38. R. Wilson (Caltech)
17. L. Haddad (U de M) 39. P. Winkler (Emory)
18. J. Hemmeter (Delaware) 40. Q. Zheng (Shangai)
19. T. Ito (Joetsu) 41. H. Zhou (Simon Fraser)
20. A.V. Ivanov (Moscow)
21. K.W. Johnson (Penn. State)
22. G. Kalai (Hebrew U.)