It was recently proved in Alon et al. ( 2017 ) that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following ordered matrix removal lemma: For any finite alphabet Γ, any hereditary property P $\mathcal {P}$ of matrices over Γ, and any 𝜖 > 0, there exists f P ( 𝜖 ) $f_{\mathcal {P}}(\epsilon )$ such that for any matrix M over Γ that is 𝜖 -far from satisfying P $\mathcal {P}$ , most of the f P ( 𝜖 ) × f P ( 𝜖 ) $f_{\mathcal {P}}(\epsilon ) \times f_{\mathcal {P}}(\epsilon )$ submatrices of M do not satisfy P $\mathcal {P}$ . Here being 𝜖 -far from P $\mathcal {P}$ means that one needs to modify at least an 𝜖 -fraction of the entries of M to make it satisfy P $\mathcal {P}$ . However, in the above general removal lemma, f P ( 𝜖 ) $f_{\mathcal {P}}(\epsilon )$ grows very quickly as a function of 𝜖 − 1 , even when P $\mathcal {P}$ is characterized by a single forbidden submatrix. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following, which can be seen as an efficient binary matrix analogue of the triangle removal lemma: For any fixed s × t binary matrix A and any 𝜖 > 0 there exists δ > 0 polynomial in 𝜖 , such that for any binary matrix M in which less than a δ -fraction of the s × t submatrices are equal to A , there exists a set of less than an 𝜖 -fraction of the entries of M that intersects every copy of A in M . We generalize the work of Alon et al. ( 2007 ) and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.

中文翻译：

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矩阵的有效去除引理

最近在 Alon 等人中证明了这一点。( 2017 ) 通过建立以下有序矩阵去除引理，有限字母表上的二维矩阵（其中不忽略行和列顺序）的任何遗传属性都可以用恒定数量的查询进行测试：对于任何有限字母表 Γ , Γ 上矩阵的任何遗传性质 P $\mathcal {P}$，并且任何 𝜖 > 0，都存在 f P ( 𝜖 ) $f_{\mathcal {P}}(\epsilon )$ 使得对于任何矩阵 M Γ 是 𝜖 - 远不能满足 P $\mathcal {P}$ ，大部分 f P ( 𝜖 ) × f P ( 𝜖 ) $f_{\mathcal {P}}(\epsilon ) \times f_{\ M 的 mathcal {P}}(\epsilon )$ 子矩阵不满足 P $\mathcal {P}$ 。这里是 𝜖 -远离 P $\mathcal {P}$ 意味着需要修改至少一个 𝜖 - 部分 M 的条目以使其满足 P $\mathcal {P}$ 。然而，在上述一般去除引理中，f P ( 𝜖 ) $f_{\mathcal {P}}(\epsilon )$ 作为 𝜖 − 1 的函数增长非常快，即使 P $\mathcal {P}$ 的特征是单个禁止子矩阵。在这项工作中，我们为上述问题的几个特殊情况建立了更有效的去除引理。特别地，我们展示了以下内容，可以将其视为三角形去除引理的有效二进制矩阵模拟：对于任何固定的 s × t 二进制矩阵 A 和任何 𝜖 > 0 存在 δ > 0 多项式在 𝜖 中，使得对于任何二进制矩阵 M，其中 s × t 子矩阵的小于 δ 分数等于 A ，存在一组小于 𝜖 的 M 条目的分数，它与 M 中 A 的每个副本相交。我们概括了 Alon 等人的工作。( 2007 ) 并在证明他们的一个猜想方面取得进展。