Odd Ramsey numbers of multipartite graphs and hypergraphs

TL;DR

This paper investigates the odd Ramsey numbers for multipartite graphs and hypergraphs, providing asymptotic results that extend previous work and are the first of their kind for hypergraphs.

Contribution

It generalizes known results to hypergraphs and multipartite graphs, establishing new asymptotic formulas for odd Ramsey numbers in these contexts.

Findings

Proves $r_{odd} (K_{n,n}, K_{2,t})=\frac{n}{t} + o(n)$ for all $t\geq 2$.

Establishes $r_{odd} (\mathcal{K}^{(k)}_{n,\dots,n}, \mathcal{K}_{1,\dots,1,2,2})=\frac{n}{2} + o(n)$ for all $k\geq 2$.

First results on odd Ramsey numbers for hypergraphs.

Abstract

Given a hypergraph $G$ and a subhypergraph $H$ of $G$, the \emph{odd Ramsey number} $r_{odd}(G,H)$ is the minimum number of colors needed to edge-color $G$ so that every copy of $H$ intersects some color class in an odd number of edges. Generalizing a result of \cite{BHZ} in two different ways, in this paper we prove $r_{odd} \left(K_{n,n}, K_{2,t} \right)=\frac{n}{t} + o(n)$ for all $t\geq 2$, and $r_{odd} \left(\mathcal{K}^{(k)}_{n,\dots,n}, \mathcal{K}_{1,\dots,1,2,2} \right) = \frac{n}{2} + o(n)$ for all $k\geq 2$. The latter is the first result studying odd Ramsey numbers for hypergraphs.