Partition Universality for Hypergraphs of Bounded Degeneracy and Degree

Abstract

We consider the following question. When is the random $k$-uniform hypergraph $\Gamma=G^{(k)}(N,p)$ likely to be $r$-partition universal for $k$-uniform hypergraphs of bounded degree and degeneracy? That is, for which~$p$ can we guarantee asymptotically almost surely that in any $r$-colouring of $E(\Gamma)$ there exists a colour $\chi$ such that in $\Gamma$ there are $\chi$-monochromatic copies of all $k$-uniform hypergraphs of maximum vertex degree $\Delta$, degeneracy at most $D$, and $cN$ vertices for some constant $c=c(D,\Delta)>0$. We show that if $\mu>0$ is fixed, then $p\ge N^{-1/D+\mu}$ suffices for a positive answer if $N$ is large. On the other hand, for $p=o(N^{-1/D})$ we show that $G^{(k)}(N,p)$ is likely not to contain some graphs of maximum degree $\Delta$ and degeneracy $D$ on $cN$ vertices at all. This improves the best upper bounds on the minimum number of edges required for a $k$-uniform hypergraph to be partition universal (even for $k=2$) and also for the size-Ramsey problem for most $k$-uniform hypergraphs of bounded degree and degeneracy.