The Existence of Exactlym-Coloured Complete Subgraphs

Given a graphG, its edges are said to be exactlyx-coloured if we have a surjective map from the edges to some set of colours of sizex. Erickson considered the following statement which he denotedP(c, m): if the edges ofKω—the complete graph on vertex set N—are exactlyc-coloured, then there exists an infinite complete subgraph ofKωwhose edges are exactlym-coloured. Ramseyʹs Theorem states thatP(c, m) is true form=1 and allc⩾1, and can easily be used to show thatP(c, m) holds whenm=2 andc⩾2. Erickson conjectured thatP(c, m) is false wheneverc>m⩾3. We prove that givenm⩾3 there exists an integerC(m) such thatP(c, m) is false for allc⩾C(m).