A Ramsey variant of the Brown-Erd\H{o}s-S\'os conjecture

TL;DR

This paper proves a Ramsey relaxation of the Brown-Erdős-Sós conjecture for linear hypergraphs, showing that in certain colorings, one can find monochromatic edges with bounded vertex span, under specific conditions on r and c.

Contribution

It establishes the Ramsey version of the conjecture for linear hypergraphs when r is sufficiently large relative to the number of colors, advancing understanding of hypergraph Ramsey properties.

Findings

Proves the Ramsey relaxation for r ≥ r_0(c)

Shows the result holds for all r ≥ 4 when c=2

Advances the study of hypergraph Ramsey theory

Abstract

An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-Erd\H{o}s-S\'os conjecture states that for every fixed $k$ and $r$, every linear $r$-graph with $\Omega(n^2)$ edges contains $k$ edges spanned by at most $(r-2)k+3$ vertices. As an intermediate step towards this conjecture, Conlon and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely, that for every fixed $k$, $r$ and $c$, in every $c$-colouring of a complete linear $r$-graph, one can find $k$ monochromatic edges spanned by at most $(r-2)k+3$ vertices. We prove that this Ramsey version of the conjecture holds under the additional assumption that $r \geq r_0(c)$, and we show that for $c=2$ it holds for all $r\geq 4$.