Monochromatic bounded degree subgraph partitions

Abstract

Let $\mathcal{F}=\{F_1,F_2,\dots \}$ be a sequence of graphs such that $F_n$ is a graph on $n$ vertices with maximum degree at most $\Delta$. We show that there exists an absolute constant $C$ such that the vertices of any 2-edge-colored complete graph can be partitioned into at most $2^{C\Delta \log \Delta }$ vertex disjoint monochromatic copies of graphs from $\mathcal{F}$. If each $F_n$ is bipartite, then we can improve this bound to $2^{C\Delta }$; this result is optimal up to the constant $C$.

References in corpus (1)

• A quantitative version of the Blow-up Lemma