Generalized Ramsey numbers of cycles, paths, and hypergraphs

TL;DR

This paper investigates bounds on generalized Ramsey numbers for hypergraphs, focusing on cycles, paths, and cliques, providing asymptotically sharp estimates for specific complete and bipartite hypergraph cases.

Contribution

It establishes new bounds, some asymptotically sharp, on generalized Ramsey numbers for hypergraphs involving cycles, paths, and cliques, expanding understanding in hypergraph Ramsey theory.

Findings

Derived bounds for hypergraph Ramsey numbers involving cycles and paths.

Provided asymptotically sharp estimates in specific hypergraph configurations.

Extended classical Ramsey results to generalized hypergraph settings.

Abstract

Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in \mathcal{H}$ in $G$ receives at least $q$ different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when $G=K_n$ or $G=K_{n,n}$ and $\mathcal{H}$ is a set of cycles or paths, and when $G=K_n^k$ and $\mathcal{H}$ contains a clique on $k+2$ vertices or a tight cycle.