Chromatic Ramsey theory (Q5961465)

If \(G\) is a countable graph which has arbitrarily large cliques and the \(k\)-tuples of \(G\) are colored with a finite number of colors then there is an infinite chromatic subgraph on which the \(k\)-tuples get \(2^{k-1}\) colors. This is sharp if \(G\) does not contain infinite cliques. There is a countable triangle-free graph such that if the pairs are colored with finitely many colors then some 3 colors cover an infinite chromatic subgraph. There is a countable triangle-free graph which has for every finite \(k\) a \(k\)-coloring of the pairs so that the vertex set of every infinite chromatic subgraph uses all colors. The homogeneous countable \(n\)-clique free graph has for every finite \(k\) a \(k\)-coloring of the pairs of vertices such that infinite chromatic subsets necessarily use at least 3 colors.