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A note on the Brown–Erdős–Sós conjecture in groups

Part of: Graph theory

Published online by Cambridge University Press:  03 February 2020

Jason Long
Jason Long*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK Email: [email protected]
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Abstract

We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some k, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with t triples spanning $ O(\sqrt t )$ vertices, which is the best possible up to the implied constant. We confirm that for all t we can find a collection of t triples spanning at most t + 3 vertices, resolving the Brown–Erdős–Sós conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemerédi’s theorem and the density Hales–Jewett theorem.

This result was discovered simultaneously and independently by Nenadov, Sudakov and Tyomkyn [5], and a weaker result avoiding the arithmetic machinery was obtained independently by Wong [11].

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 05C35: Extremal problems 05C65: Hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 4 , July 2020 , pp. 633 - 640
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Copyright
© Cambridge University Press 2020

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References

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