Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T01:07:51.883Z Has data issue: false hasContentIssue false
  • English
  • Français

The (7, 4)-Conjecture in Finite Groups

Part of: Extremal combinatorics Algebraic combinatorics

Published online by Cambridge University Press:  10 February 2015

JÓZSEF SOLYMOSI
JÓZSEF SOLYMOSI*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z4, Canada (e-mail: solymosi@math.ubc.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

The first open case of the Brown–Erdős–Sós conjecture is equivalent to the following: for every c > 0, there is a threshold n0 such that if a quasigroup has order n ⩾ n0, then for every subset S of triples of the form (a, b, ab) with |S| ⩾ cn2, there is a seven-element subset of the quasigroup which spans at least four triples of S. In this paper we prove the conjecture for finite groups.

MSC classification

Primary: 05D05: Extremal set theory
Secondary: 05E15: Combinatorial aspects of groups and algebras
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 4: Oberwolfach Special Issue Part 1 , July 2015 , pp. 680 - 686
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ajtai, M. and Szemerédi, E. (1974) Sets of lattice points that form no squares. Stud. Sci. Math. Hungar. 9 9–11.Google Scholar
[2]Brown, W. G., Erdős, P. and Sós, V. T. (1973) On the existence of triangulated spheres in 3-graphs and related problems. Period. Math. Hungar. 3 221–228.Google Scholar
[3]Bryant, D. and Horsley, D. (2009) A proof of Lindner's conjecture on embeddings of partial Steiner triple systems. J. Combin. Des. 17 63–89.CrossRefGoogle Scholar
[4]Erdős, P. and Straus, E. G. (1975/76) How abelian is a finite group? Linear and Multilinear Algebra 3 307–312.CrossRefGoogle Scholar
[5]Forbes, A. D., Grannell, M. J. and Griggs, T. S.(2007) On 6-sparse Steiner triple systems. J. Combin. Theory Ser. A 114 235–252.CrossRefGoogle Scholar
[6]Frankl, P. and Rödl, V. (2002) Extremal problems on set systems. Random Struct. Alg. 20 131–164.CrossRefGoogle Scholar
[7]Fujiwara, Y. (2006) Sparseness of triple systems: A survey. RIMS Kokyuroku 1465 173–185.Google Scholar
[8]Gowers, W. T. (2007) Hypergraph regularity and the multidimensional Szemerédi theorem. Ann. of Math. 166 897–946.CrossRefGoogle Scholar
[9]Lindner, C. C. (1975) A partial Steiner triple system of order n can be embedded in a Steiner triple system of order 6n + 3. J. Combin. Theory Ser. A 18 349–351.CrossRefGoogle Scholar
[10]McKay, B., Meynert, A. and Myrvold, W. (2007) Small Latin squares, quasigroups, and loops. J. Combin. Des. 15 98–119.CrossRefGoogle Scholar
[11]Miller, G. A., Blichfeldt, H. F. and Dickson, L. E. (1916) Theory and Applications of Finite Groups, Wiley.Google Scholar
[12]Nagle, B., Rödl, V. and Schacht, M. (2006) The counting lemma for regular k-uniform hypergraphs. Random Struct. Alg. 28 113–179.CrossRefGoogle Scholar
[13]Pyber, L. (1997) How abelian is a finite group? In The Mathematics of Paul Erdős I, Vol. 13 of Algorithms and Combinatorics, Springer, pp. 372–384.Google Scholar
[14]Rödl, V. and Skokan, J. (2004) Regularity lemma for k-uniform hypergraphs. Random Struct. Alg. 25 1–42.CrossRefGoogle Scholar
[15]Ruzsa, I. Z. and Szemerédi, E. (1978) Triple systems with no six points carrying three triangles. In Combinatorics: Proc. Fifth Hungarian Colloq., Vol. II (Keszthely 1976), Vol. 18 of Colloquia Mathematica Societatis János Bolyai, North-Holland, pp. 939–945.Google Scholar
[16]Solymosi, J. (2004) A note on a question of Erdős and Graham. Combin. Probab. Comput. 13 263–267.CrossRefGoogle Scholar
[17]Solymosi, J. (2013) Roth-type theorems in finite groups. European J. Combin. 34 1454–1458.CrossRefGoogle Scholar
[18]Szemerédi, E. (1978) Regular partitions of graphs. Problèmes Combinatoires et Théorie des Graphes: Colloq. Internat. CNRS, Univ. Orsay (Orsay 1976), Vol. 260 of Colloq. Internat. CNRS, CNRS, pp. 399–401.Google Scholar