Odd Ramsey numbers of multipartite graphs and hypergraphs

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 $r_{odd}(K_{n,n},K_{2,t})=\frac{n}{t}+o(n)$ for all $t\ge 2$, and $r_{odd}({\mathcal{K}}_{n,\dots ,n}^{(k)},{\mathcal{K}}_{1,\dots ,1,2,2})=\frac{n}{2}+o(n)$ for all $k\ge 2$. The latter is the first result studying odd Ramsey numbers for hypergraphs.