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arXiv · 2405.15904

Generalized Ramsey numbers of cycles, paths, and hypergraphs

Deepak Bal·Patrick Bennett·Emily Heath·Shira Zerbib
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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.

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Deepak Bal, Patrick Bennett, Emily Heath, Shira Zerbib. 2024-05-24. Generalized Ramsey numbers of cycles, paths, and hypergraphs. https://arxiv.org/abs/2405.15904

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