Monochromatic bounded degree subgraph partitions

Abstract

Let ${\cal{F}}=\{F_1,F_2,\ldots\}$ 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 ${\cal{F}}$. If each $F_n$ is bipartite, then we can improve this bound to $2^{C \Delta}$; this result is optimal up to the constant $C$.