Multiplicities of subgraphs

A former conjecture of Burr and Rosta [1], extending a conjecture of Erdős [2], asserted that in any two-colouring of the edges of a large complete graph, the proportion of subgraphs isomorphic to a fixed graphG which are monochromatic is at least the proportion found in a random colouring. It is now known that the conjecture fails for some graphsG, includingG=Kp forp≥4.We investigate for which graphsG the conjecture holds. Our main result is that the conjecture fails ifG containsK4 as a subgraph, and in particular it fails for almost all graphs.