A Ramsey variant of the Brown–Erdős–Sós conjecture

Asaf Shapira, Mykhaylo Tyomkyn

Research output: Contribution to journalArticlepeer-review

Abstract

An (Formula presented.) -uniform hypergraph ((Formula presented.) -graph for short) is called linear if every pair of vertices belongs to at most one edge. A linear (Formula presented.) -graph is complete if every pair of vertices is in exactly one edge. The famous Brown–Erdős–Sós conjecture states that for every fixed (Formula presented.) and (Formula presented.), every linear (Formula presented.) -graph with (Formula presented.) edges contains (Formula presented.) edges spanned by at most (Formula presented.) vertices. As an intermediate step towards this conjecture, Conlon and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely, that for every fixed (Formula presented.), (Formula presented.) and (Formula presented.), in every (Formula presented.) -colouring of a complete linear (Formula presented.) -graph, one can find (Formula presented.) monochromatic edges spanned by at most (Formula presented.) vertices. We prove that this Ramsey version of the conjecture holds under the additional assumption that (Formula presented.), and we show that for (Formula presented.) it holds for all (Formula presented.).

Original languageEnglish
Pages (from-to)1453-1469
Number of pages17
JournalBulletin of the London Mathematical Society
Volume53
Issue number5
DOIs
StatePublished - Oct 2021

Keywords

  • 05C15
  • 05C65 (primary)

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