Off-diagonal Ramsey numbers for slowly growing hypergraphs

TL;DR

This paper establishes lower bounds on off-diagonal Ramsey numbers for certain slowly growing hypergraphs, revealing new growth rates and employing advanced combinatorial techniques.

Contribution

It proves that for non $k$-partite slowly growing hypergraphs, the Ramsey number grows at least as fast as $n^k$ over polylogarithmic factors, a novel lower bound.

Findings

Established lower bounds for off-diagonal Ramsey numbers of slowly growing hypergraphs.

Identified the order of the Ramsey number for the specific hypergraph $F_5$ as $n^3$ over polylogarithmic factors.

Used pseudorandom graphs, martingales, and hypergraph containers in the constructions.

Abstract

For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A hypergraph is $\textit{slowly growing}$ if there is an ordering $e_1,e_2,\dots,e_t$ of its edges such that $|e_i \setminus \bigcup_{j = 1}^{i - 1}e_j| \leq 1$ for each $i \in \{2, \ldots, t\}$. We prove that if $k \geq 3$ is fixed and $F$ is any non $k$-partite slowly growing $k$-uniform hypergraph, then for $n\ge2$, \[ r(F,n) = \Omega\Bigl(\frac{n^k}{(\log n)^{2k - 2}}\Bigr).\] In particular, we deduce that the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^{3}/\mbox{polylog}(n)$, where $F_5$ is the triple system $\{123, 124, 345\}$. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers.