Zarankiewicz problem on arvau

Wikipedia / Zarankiewicz problem

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 a set of edges. No two edges can both connect the same pair of vertices. A complete bipartite graph that are all connected to each other, then these vertices induce, not vice versa.)

The Zarankiewicz function. The Zarankiewicz problem asks for a formula for the Zarankiewicz function, or (failing that) for tight asymptotic bounds goes to infinity.

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 rectangle in the integer lattice grid. such that every (0,1)-matrix submatrix is made up only of 1s.

Examples

vertices on each side that has no 4-cycle (its girth (a hexagon).

. The answers were supplied soon afterwards by Wacław Sierpiński. 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 subgraph.

Upper bounds

The Kővári–Sós–Turán theorem provides an upper bound on the solution to the Zarankiewicz problem. It was established by Tamás Kővári, Vera T. Sós and Pál Turán shortly after the problem had been posed:

. Shortly afterwards, Hyltén-Cavallius observed that essentially the same argument can be used to prove the above inequality. was given by Štefan Znám:

are assumed to be constant, then asymptotically, using the big O notation, these formulae can be expressed as

;
.

, assuming without loss of generality, we have the asymptotic upper bound

Lower bounds

, we have

, by taking its bipartite double cover., it has been conjectured that

), the above statements have been proved using various algebraic and random algebraic constructions. At the same time, the answer to the general question is still unknown to us.

Incidence graphs in finite geometry

may be obtained as the Levi graph, or point-line incidence graph, of a projective plane edges. Since this lower bound matches the upper bound given by I. Reiman, we have the asymptotic

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.

. Hence we have the upper bound

Norm graphs and projective norm graphs

was verified by Kollár, Rónyai, and Szabó and Alon, Rónyai, and Szabó using the construction of norm graphs and projective norm graphs over finite fields.

, consider the norm graph NormGraphp,s is the norm map

must satisfy

solutions.

using the projective norm graph, a construction slightly stronger than the above. The projective norm graph ProjNormGraphp,s edges.

., we have

common neighbors.

Clique partitions

Using a related result on clique partition numbers, Alon, Mellinger, Mubayi and Verstraëte , then we have

is a clique partition.

Randomized algebraic constructions

were also given by Blagojević, Bukh and Karasev and by Bukh .

. Let

, which implies

, using similar ideas but with more tools from algebraic geometry.

Applications

The Kővári–Sós–Turán theorem has been used in discrete geometry lines in the Euclidean plane are nearly equal, in which case the Szemerédi–Trotter theorem 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.