K\H{o}v\'ari-S\'os-Tur\'an theorem for hereditary families

The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose the systematic study of this phenomenon, and state the conjecture that if $H$ is a bipartite graph, then an induced $H$-free and $K_{s,s}$-free graph cannot have much more edges than an $H$-free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by $k$ on one side, obtaining in all the cases similar bounds as in the non-induced setting. Our results also have applications to the Erd\H{o}s-Hajnal conjecture, the problem of finding induced $C_4$-free subgraphs with large degree and bounding the average degree of $K_{s, s}$-free graphs which do not contain induced subdivisions of a fixed graph.