The binding number of a graph and its cliques

We consider the binding numbers of KrKr-free graphs, and improve the upper bounds on the binding number which force a graph to contain a clique of order rr. For the case r=4r=4, we provide a construction for K4K4-free graphs which have a larger binding number than the previously known constructions. This leads to a counterexample to a conjecture by Caro regarding the neighborhoods of independent sets.