On the size-Ramsey number of hypergraphs

TL;DR

This paper introduces the concept of size-Ramsey numbers for hypergraphs, exploring their properties and highlighting the complexity and open problems in determining these numbers for various hypergraph classes.

Contribution

It extends the study of size-Ramsey numbers from graphs to hypergraphs, initiating the investigation for k-uniform hypergraphs and analyzing specific classes like cliques and paths.

Findings

Size-Ramsey numbers for hypergraphs are highly complex to determine.

Many open problems remain in understanding hypergraph size-Ramsey numbers.

Abstract

The size-Ramsey number of a graph $G$ is the minimum number of edges in a graph $H$ such that every 2-edge-coloring of $H$ yields a monochromatic copy of $G$. Size-Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size-Ramsey numbers for $k$-uniform hypergraphs. Analogous to the graph case, we consider the size-Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size-Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.