# On Berge-Ramsey problems

## TL;DR

This paper investigates the Ramsey numbers for Berge copies of graphs within hypergraphs, establishing conditions for super-linear growth, improving lower bounds, and providing sharp bounds for specific cases.

## Contribution

It introduces new bounds and methods for Berge-Ramsey problems, including conditions for super-linear Ramsey numbers and sharp bounds for certain parameters.

## Key findings

- Identified when Ramsey numbers for Berge copies can be super-linear.
- Developed a new approach to lower bounds, significantly improving previous results.
- Provided a nearly sharp upper bound for the case G=K_n and r=2c-1.

## Abstract

Given a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ if $V(G)\subset V(\mathcal{H})$ and there is a bijection $f:E(G)\rightarrow E(\mathcal{H})$ such that for any edge $e$ of $G$ we have $e\subset f(e)$. We study Ramsey problems for Berge copies of graphs, i.e. the smallest number of vertices of a complete $r$-uniform hypergraph, such that if we color the hyperedges with $c$ colors, there is a monochromatic Berge copy of $G$. We obtain a couple results regarding these problems. In particular, we determine for which $r$ and $c$ the Ramsey number can be super-linear. We also show a new way to obtain lower bounds, and improve the general lower bounds by a large margin. In the specific case $G=K_n$ and $r=2c-1$, we obtain an upper bound that is sharp besides a constant term, improving earlier results.