A conjecture concerning Ramsey's theorem (Q1318831)

An exact \(c\)-coloring of the edges of a graph is defined to be one in which all \(c\) colors are used. Ramsey's Theorem is generalized accordingly: let \(P(c,m)\) be the statement that every exact \(c\)-coloring of the edges of a countable infinite complete graph yields an exactly \(m\)-colored infinite complete subgraph. It is inquired as to which ordered pairs \((c,m)\) make \(P(c,m)\) true. Sufficient conditions are found, and it is conjectured that these conditions are necessary. Constructions show that this is true for a density 1 of ordered pairs \((c,m)\).