A note on the Brown--Erdős--Sós conjecture in groups
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by
Jason Long
2019
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(√(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.
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1902.07693v3
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