Forbidden Rainbow Subgraphs That Force Large Highly Connected Monochromatic Subgraphs

Abstract

We consider a forbidden rainbow structure condition which implies that an edge colored complete graph has an almost spanning monochromatic subgraph with high connectivity. Namely, we classify the connected graphs $G$ that satisfy the following statement: If $n\,{\gg}\,m\,{\gg}\,k$ are integers, then any rainbow $G$-free coloring of the edges of $K_{n}$ using $m$ colors contains a monochromatic $k$-connected subgraph of order at least $n - f(G, k, m)$, where $f$ does not depend on $n$.