On the Zarankiewicz Problem for Graphs with Bounded VC-Dimension

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph \(K_{k,k}\) as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is \(O\left( n^{2 - 1/k}\right) \). An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most d, where d is a fixed integer such that \(k \ge d \ge 2\). A remarkable result of Fox et al. (J. Eur. Math. Soc. (JEMS) 19:1785–1810, 2017) with multiple applications in incidence geometry shows that, under this additional hypothesis, the number of edges in a bipartite graph on n vertices and with no copy of \(K_{k,k}\) as a subgraph must be \(O\left( n^{2 - 1/d}\right) \). This theorem is sharp when \(k=d=2\), because by design any \(K_{2,2}\)-free graph automatically has VC-dimension at most 2, and there are well-known examples of such graphs with \(\Omega \left( n^{3/2}\right) \) edges. However, it turns out this phenomenon no longer carries through for any larger d. We show the following improved result: the maximum number of edges in bipartite graphs with no copies of \(K_{k,k}\) and VC-dimension at most d is \(o(n^{2-1/d})\), for every \(k \ge d \ge 3\).