Zarankiewicz problem

Problem statement

A bipartite graph G = (U, V, E) consists of two disjoint sets of vertices U and V, and a set of edges each of which connects a vertex in U to a vertex in V. 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 U and a vertex from V is connected to each other. A complete bipartite graph in which U has s vertices and V has t vertices is denoted K_s,t. If G = (U, V, E) is a bipartite graph, and there exists a set of s vertices of U and t vertices of V that are all connected to each other, then these vertices induce a subgraph of the form K_s,t. (In this formulation, the ordering of s and t is significant: the set of s vertices must be from U and the set of t vertices must be from V, not vice versa.)

The Zarankiewicz function z(m, n; s, t) denotes the maximum possible number of edges in a bipartite graph G = (U, V, E) for which |U| = m and |V| = n, but which does not contain a subgraph of the form K_s,t. As a shorthand for an important special case, z(n; t) is the same as z(n, n; t, t). The Zarankiewicz problem asks for a formula for the Zarankiewicz function, or (failing that) for tight asymptotic bounds on the growth rate of z(n; t) assuming that t is a fixed constant, in the limit as n goes to infinity.

For s = t = 2 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 G = (U, V, E) can be visualized as the points of a |U| × |V| 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, z(m, n; s, t) denotes the maximum number of points that can be placed within an m × n grid in such a way that no subset of rows and columns forms a complete s × t grid. An alternative and equivalent definition is that z(m, n; s, t) is the smallest integer k such that every (0,1)-matrix of size m × n with k + 1 ones must have a set of s rows and t columns such that the corresponding s×t submatrix is made up only of 1's.

Examples

The number z(n, 2) asks for the maximum number of edges in a bipartite graph with n vertices on each side that has no 4-cycle (its girth is six or more). Thus, z(2, 2) = 3 (achieved by a three-edge path), and z(3, 2) = 6 (a hexagon).

In his original formulation of the problem, Zarankiewicz asked for the values of z(n; 3) for n = 4, 5, and 6. The answers were supplied soon afterwards by Wacław Sierpiński: z(4; 3) = 13, z(5; 3) = 20, and z(6; 3) = 26. The case of z(4; 3) is relatively simple: a 13-edge bipartite graph with four vertices on each side of the bipartition, and no K_3,3 subgraph, may be obtained by adding one of the long diagonals to the graph of a cube. In the other direction, if a bipartite graph with 14 edges has four vertices on each side, then two vertices on each side must have degree four. Removing these four vertices and their 12 incident edges leaves a nonempty set of edges, any of which together with the four removed vertices forms a K_3,3 subgraph.

Upper bounds

The following upper bound was established by Tamás Kővári, Vera T. Sós and Pál Turán shortly after the problem had been posed, and has become known as the Kővári–Sós–Turán theorem:

In fact, Kővári, Sós, and Turán proved a similar inequality for z(n; t), but shortly afterwards, Hyltén-Cavallius observed that essentially the same argument can be used to prove the above inequality. An improvement to the constant factor in the second term of this formula, in the case of z(n; t), was given by Štefan Znám:

If s and t are assumed to be constant, then asymptotically, using big O notation, these formulas can be expressed as

and

Lower bounds

For t = 2, and for infinitely many values of n, a bipartite graph with n vertices on each side, Ω(n^3/2) edges, and no K_2,2 may be obtained as the Levi graph of a finite projective plane, a system of n points and lines in which each two points belong to a unique line and each two lines intersect in a unique point. The graph formed from this geometry has a vertex on one side of its bipartition for each point, a vertex on the other side of its bipartition for each line, and an edge for each incidence between a point and a line. The projective planes defined from finite fields of order p lead to K_2,2-free graphs with n = p² + p + 1 and with (p² + p + 1)(p + 1) edges. For instance, the Levi graph of the Fano plane gives rise to the Heawood graph, a bipartite graph with seven vertices on each side, 21 edges, and no 4-cycles, showing that z(7; 2) ≥ 21. The lower bound on the Zarankiewicz function given by this family of examples matches an upper bound given by I. Reiman. Thus, for t = 2 and for those values of n for which this construction can be performed, it provides a precise answer to the Zarankiewicz problem. For other values of n, it follows from these upper and lower bounds that asymptotically

More generally,

For t = 3, and for infinitely many values of n, bipartite graphs with n vertices on each side, Ω(n^5/3) edges, and no K_3,3 may again be constructed from finite geometry, by letting the vertices represent points and spheres (of a carefully chosen fixed radius) in a three-dimensional finite affine space, and letting the edges represent point-sphere incidences.

It has been conjectured that

for all constant values of t, but this is only known for t = 2 and t = 3 by the above constructions. Tight bounds are also known for pairs (s, t) with widely differing sizes (specifically s ≥ (t − 1)!). For such pairs,

lending support to the above conjecture.

Non-bipartite graphs

Up to constant factors, z(n; t) also bounds the number of edges in an n-vertex graph (not required to be bipartite) that has no K_t,t subgraph. For, in one direction, a bipartite graph with z(n; t) edges and with n vertices on each side of its bipartition can be reduced to a graph with n vertices and (in expectation) z(n; t)/4 edges, by choosing n/2 vertices uniformly at random from each side. In the other direction, a graph with n vertices and no K_t,t can be transformed into a bipartite graph with n vertices on each side of its bipartition, twice as many edges, and still no K_t,t by taking its bipartite double cover.

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 n points and m lines in the Euclidean plane necessarily has no K_2,2, so by the Kővári–Sós–Turán it has O(nm^1/2 + m) point-line incidences. This bound is tight when m is much larger than n, but not when m and n are nearly equal, in which case the Szemerédi–Trotter theorem provides a tighter O(n^2/3m^2/3 + n + m) 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.