Bipartite Turán Problems for Ordered Graphs

Abstract

A zero-one matrix M contains a zero-one matrix A if one can delete some rows and columns of M, and turn some 1-entries into 0-entries such that the resulting matrix is A. The extremal number of A, denoted by ex(n, A), is the maximum number of 1-entries in an n × n sized matrix M that does not contain A.

A matrix A is column-t-partite (or row-t-partite), if it can be cut along the columns (or rows) into t submatrices such that every row (or column) of these submatrices contains at most one 1-entry. We prove that if A is column-t-partite, then ${\rm{ex}}(n,A) < {n^{2 - {1 \over t} + {1 \over {{t^2}}} + o(1)}}$ and if A is both column- and row-t-partite, then ${\rm{ex}}(n,A) < {n^{2 - {1 \over t} + o(1)}}$. Our proof combines a novel density-increment-type argument with the celebrated dependent random choice method.

Results about the extremal numbers of zero-one matrices translate into results about the Turán numbers of bipartite ordered graphs. In particular, a zero-one matrix with at most t 1-entries in each row corresponds to a bipartite ordered graph with maximum degree t in one of its vertex classes. Our results are partially motivated by a well-known result of Füredi (1991) and Alon, Krivelevich, Sudakov (2003) stating that if H is a bipartite graph with maximum degree t in one of the vertex classes, then ${\rm{ex}}(n,H) = O({n^{2 - {1 \over t}}})$. The aim of the present paper is to establish similar general results about the extremal numbers of ordered graphs.