Subgraph densities in Kr-free graphs

A counterexample to a recent conjecture of LidickATIN SMALL LETTER Y WITH ACUTE and Murphy on the structure of Kr-free graph maximizing the number of copies of a given graph with chromatic number at most r -1 is known in the case r = 3. Here, we show that this conjecture does not hold for any r, and that the structure of extremal graphs can be richer. We also provide an alternative conjecture and, as a step towards its proof, we prove an asymptotically tight bound on the number of copies of any bipartite graph of radius at most 2 in the class of triangle-free graphs.Mathematics Subject Classifications: 05C35