Variants of the Erdos-Szekeres and Erdos-Hajnal Ramsey problems

TL;DR

This paper investigates the growth rates of Ramsey numbers for powers of paths and hypergraphs, providing new bounds and conjectures for ordered Ramsey problems, advancing understanding of combinatorial structures.

Contribution

It improves upper bounds for the Ramsey numbers of path powers, determines the tower growth rate for hypergraph Ramsey numbers, and proposes conjectures for ordered hypergraph Ramsey towers.

Findings

Proved $r(P_n^{2}, P_n^{2})< cn^{19.5}$, improving previous bounds.

Determined the tower growth rate for hypergraph Ramsey numbers involving cliques and paths.

Conjectured the optimal tower height for an ordered Erdős-Hajnal hypergraph Ramsey problem.

Abstract

Given integers $\ell,n$, the $\ell$th power of the path $P_n$ is the ordered graph $P_n^{\ell}$ with vertex set $v_1<v_2<\cdots < v_n$, and all edges of the form $v_iv_j$ where $|i-j|\le \ell$. The ramsey number $r(P_n^{\ell}, P_n^{\ell})$ is the minimum $N$ such that every 2-coloring of ${[N] \choose 2}$ results in a monochromatic copy of $P_n^{\ell}$. It is well-known that $r(P_n^1, P_n^1)=(n-1)^2+1$. For $\ell>1$, Balko-Cibulka-Kr\'al-Kyn\v{c}l proved that $r(P_n^{\ell}, P_n^{\ell})< c_{\ell}n^{128 \ell}$ and asked for the growth rate for fixed $\ell$. When $\ell=2$, we improve this upper bound by proving $r(P_n^{2}, P_n^{2})< cn^{19.5}$. Using this result, we determine the correct tower growth rate of the $k$-uniform hypergraph ramsey number of a $(k+1)$-clique versus an ordered tight path. Finally, we consider an ordered version of the classical Erd Hos-Hajnal hypergraph ramsey problem, improve the tower height given by the trivial upper bound, and conjecture that this tower height is optimal.