2019/02/20 by Jason Long, Long, Jason
Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1902.07693
openalex publication_date 2019/02/20 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
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--Erd\H os--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.