Sparse multipartite graphs as partition universal for graphs with bounded degree

Abstract

For graphs G and H, let $G\rightarrow (H,H)$ signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Denote $\mathcal {H}(\Delta ,n)=\{H:|V(H)|=n,\Delta (H)\le \Delta \}$. For any $\Delta$ and n, we say that G is partition universal for $\mathcal {H}(\Delta ,n)$ if $G\rightarrow (H,H)$ for every $H\in \mathcal {H}(\Delta ,n)$. Let $G_r(N,p)$ be the random spanning subgraph of the complete r-partite graph $K_r(N)$ with N vertices in each part, in which each edge of $K_r(N)$ appears with probability p independently and randomly. We prove that for fixed $\Delta \ge 2$ there exist constants r, B and C depending only on $\Delta$ such that if $N\ge Bn$ and $p=C(\log N/N)^{1/\Delta }$, then asymptotically almost surely $G_r(N,p)$ is partition universal for $\mathcal {H}(\Delta ,n)$.