A note on hypergraphs with asymmetric Ramsey properties

TL;DR

This paper constructs hypergraphs demonstrating specific asymmetric Ramsey properties, extending known results from graphs to hypergraphs and highlighting nuanced differences in Ramsey behavior.

Contribution

It proves the existence of hypergraphs with particular asymmetric Ramsey properties, generalizing prior graph results to hypergraphs for the first time.

Findings

Existence of hypergraphs not Ramsey for certain configurations

Hypergraphs can have asymmetric Ramsey properties similar to graphs

Extends recent graph results to hypergraph setting

Abstract

Let $r,\ell\geq2$ be integers. Given $r$-graphs $G$ and $F_1,\dots,F_\ell$, we write $G\to(F_1,\dots,F_\ell)$ if every $\ell$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$th color for some $1\leq i\leq\ell$, otherwise we write $G\not\to(F_1,\dots,F_\ell)$. The Ramsey number $R(F_1,\dots,F_\ell)$ is the minimum number of vertices in an $r$-graph $G$ satisfying $G\to(F_1,\dots,F_\ell)$. In this note we prove that for any integers $t_1\geq\dots\geq t_\ell>r$, there exists an $r$-graph $G$ such that $G\not\to(K^{(r)}_{t_1},\dots,K^{(r)}_{t_\ell})$ but $G\to(K^{(r)}_s,K^{(r)}_{t_\ell-1})$, where $s=R(K^{(r)}_{t_1},\dots,K^{(r)}_{t_\ell})-1$. This extends recent work by Mendon\c{c}a, Miralaei, and Mota, who established the statement for $r=2$.