A note on induced Ramsey numbers

TL;DR

This paper investigates the induced Ramsey number for hypergraphs, establishing upper bounds related to classical Ramsey numbers using the hypergraph container method, thus advancing understanding of induced monochromatic structures.

Contribution

It provides new bounds on induced Ramsey numbers of hypergraphs, showing they are controlled by classical Ramsey numbers and applying the hypergraph container method.

Findings

Induced Ramsey number is bounded by a power of the classical Ramsey number.

For 3-uniform hypergraphs, the induced Ramsey number is at most double exponential in the number of vertices.

The hypergraph container method is effective in deriving these bounds.

Abstract

The induced Ramsey number $r_{\mathrm{ind}}(F)$ of a $k$-uniform hypergraph $F$ is the smallest natural number $n$ for which there exists a $k$-uniform hypergraph $G$ on $n$ vertices such that every two-coloring of the edges of $G$ contains an induced monochromatic copy of $F$. We study this function, showing that $r_{\mathrm{ind}}(F)$ is bounded above by a reasonable power of $r(F)$. In particular, our result implies that $r_{\mathrm{ind}}(F) \leq 2^{2^{ct}}$ for any $3$-uniform hypergraph $F$ with $t$ vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on an application of the hypergraph container method.