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.

一个零一矩阵M包含一个零一矩阵A ，如果一个人可以删除M的一些行和列，并将一些 1 条目转换为 0 条目，这样得到的矩阵就是A。A的极值数，用 ex( n, A ) 表示，是不包含A的n × n大小的矩阵M中 1 项的最大数量。

一个矩阵A是column- t -partite（或 row- t -partite），如果它可以沿着列（或行）被切割成t个子矩阵，使得这些子矩阵的每一行（或列）最多包含一个 1-入口。我们证明如果A是t列分，则\({\rm{ex}}(n,A) < {n^{2 - {1 \over t} + {1 \over {{t^2 }}} + o(1)}}\)并且如果A既是列分又是行t分，则\({\rm{ex}}(n,A) < {n^{2 - {1 \over t} + o(1)}}\)。我们的证明结合了一种新颖的密度增量型论证和著名的依赖随机选择方法。

关于零一矩阵的极值数的结果转化为关于二部有序图的 Turán 数的结果。特别是，每行中最多有t个 1 项的零一矩阵对应于在其顶点类之一中具有最大度t的二分有序图。我们的结果部分受到 Füredi (1991) 和 Alon, Krivelevich, Sudakov (2003) 的一个著名结果的推动，他指出如果H是在一个顶点类中具有最大度t的二部图，则\({\rm {ex}}(n,H) = O({n^{2 - {1 \over t}}})\)。本文的目的是建立关于有序图的极值的类似的一般结果。