The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete bipartite subgraphs of a given size. It belongs to the field of extremal graph theory, a branch of combinatorics, and is named after the Polish mathematician Kazimierz Zarankiewicz, who proposed several special cases of the problem in 1951. Problem statement A bipartite graph consists of two disjoint sets of vertices and , and a set of edges each of which connects a vertex in to a vertex in . No two edges can both connect the same pair of vertices. A complete bipartite graph is a bipartite graph in which every pair of a vertex from and a vertex from is connected to each other. A complete bipartite graph in which has vertices and has vertices is denoted . If is a bipartite graph, and there exists a set of vertices of and vertices of that are all connected to each other, then these vertices induce a subgraph of the form . (In this formulation, the ordering of and is significant: the set of vertices must be from and the set of vertices must be from , not vice versa.) The Zarankiewicz function denotes the maximum possible number of edges in a bipartite graph for which and , but which does not contain a subgraph of the form . As a shorthand for an important special case, is the same as . The Zarankiewicz problem asks for a formula for the Zarankiewicz function, or (failing that) for tight asymptotic bounds on the growth rate of assuming that is a fixed constant, in the limit as goes to infinity. For this problem is the same as determining cages with girth six. The Zarankiewicz problem, cages and finite geometry are strongly interrelated. The same problem can also be formulated in terms of digital geometry. The possible edges of a bipartite graph can be visualized as the points of a rectangle in the integer lattice, and a complete subgraph is a set of rows and columns in this rectangle in which all points are present. Thus, denotes the maximum number of points that can be placed within an grid in such a way that no subset of rows and columns forms a complete grid. An alternative and equivalent definition is that is