Clique covers of H-free graphs

It takes n2/4 cliques to cover all the edges of a complete bipartite graph Kn/2,n/2, but how many cliques does it take to cover all the edges of a graph G if G has no Kt,t induced subgraph? We prove that O(n2-1/(2t)) cliques suffice for every n-vertex graph; and also prove that, even for graphs with no stable set of size four, we may need more than linearly many cliques. This settles two questions discussed at a recent conference in Lyon. (c) 2023 Published by Elsevier Ltd.