Ramsey numbers for multiple copies of hypergraphs

TL;DR

This paper determines the Ramsey numbers for large hypergraphs involving specific structures like loose paths, cycles, stars, and Kneser hypergraphs, extending known results to new classes of hypergraphs.

Contribution

It provides explicit formulas and bounds for Ramsey numbers involving multiple copies of certain hypergraphs, including new classes like linear hypergraphs and bipartite graphs with matchings.

Findings

Explicit formulas for R(G, nH) for large n

Bounds for R(mG, nH) when m ≥ n

Determination of R(G, nH) for bipartite G and arbitrary H

Abstract

In this paper, for sufficiently large $n$ we determine the Ramsey number $R(G,nH)$ where $G$ is a $k$-uniform hypergraph with the maximum independent set that intersects each of the edges in $k-1$ vertices and $H$ is a $k$-uniform hypergraph with a vertex so that the hypergraph induced by the edges containing this vertex is a star. There are several examples for such $G$ and $H$, among them are any disjoint union of $k$-uniform hypergraphs involving loose paths, loose cycles, tight paths, tight cycles with a multiple of $k$ edges, stars, Kneser hypergraphs and complete $k$-uniform $k$-partite hypergraphs for $G$ and linear hypergraphs for $H$. As an application, $R(mG,nH)$ is determined where $m$ or $n$ is large and $G$ and $H$ are either loose paths, loose cycles, tight paths, or stars. Also, $R(G,nH)$ is determined when $G$ is a bipartite graph with a matching saturating one of its color classes and $H$ is an arbitrary graph for sufficiently large $n$. Moreover, some bounds are given for $R(mG,nH)$ which allow us to determine this Ramsey number when $m\geq n$ and $G$ and $H$, $(|V(G)|\geq |V(H)|)$, are 3-uniform loose paths or cycles, $k$-uniform loose paths or cycles with at most 4 edges and $k$-uniform stars with 3 edges.