have been a number of results verifying the conjecture for different values of , using similar ideas but with more tools from algebraic geometry. Applications The Kővári–Sós–Turán theorem has been used in discrete geometry to bound the number of incidences between geometric objects of various types. As a simple example, a set of points and lines in the Euclidean plane necessarily has no , so by the Kővári–Sós–Turán it has point-line incidences. This bound is tight when is much larger than , but not when and are nearly equal, in which case the Szemerédi–Trotter theorem provides a tighter bound. However, the Szemerédi–Trotter theorem may be proven by dividing the points and lines into subsets for which the Kővári–Sós–Turán bound is tight. See also Biclique-free graph, sparse graphs whose sparsity is controlled by the solution to the Zarankiewicz problem Forbidden subgraph problem, a non-bipartite generalization of the Zarankiewicz problem Forbidden graph characterization, families of graphs defined by forbidden subgraphs of various types Turán's theorem, a bound on the number of edges of a graph with a forbidden complete subgraph References Extremal graph theory Mathematical problems Unsolved problems in graph theory Bipartite graphs