Tightness of the Kővári–Sós–Turán bound for larger complete bipartite graphs

Determine whether the Kővári–Sós–Turán upper bound O_t(n^{2-1/t}) for K_{t,t}-free n-vertex graphs is asymptotically tight for every integer t greater than 3.

Background

The paper places polygon visibility graphs within the classical Zarankiewicz problem, which asks for the maximum number of edges in a graph avoiding a prescribed complete bipartite subgraph. The Kővári–Sós–Turán theorem gives an O_t(n^{2-1/t}) upper bound for K_{t,t}-free graphs, and the paper notes that this estimate is asymptotically tight for t=2 and t=3.

For larger values of t, the authors identify the asymptotic tightness of this general extremal bound as a major unresolved problem in combinatorics. This issue is broader than the geometric bounds established in the paper and is presented as background motivation.