A $0,1$ matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers $k$, there exists a square regular $0,1$ matrix with binary rank $k$, such that the Boolean rank of its complement is $k^{\widetilde{\Omega}(\log k)}$. Equivalently, the ones in the matrix can be partitioned into $k$ combinatorial rectangles, whereas the number of rectangles needed for any cover of its zeros is $k^{\widetilde{\Omega}(\log k)}$. This settles, in a strong form, a question of Pullman (Linear Algebra Appl., 1988) and a conjecture of Hefner, Henson, Lundgren, and Maybee (Congr. Numer., 1990). The result can be viewed as a regular analogue of a recent result of Balodis, Ben-David, G\"{o}\"{o}s, Jain, and Kothari (FOCS, 2021), motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers $k$, there exists a regular graph with biclique partition number $k$ and chromatic number $k^{\widetilde{\Omega}(\log k)}$.

翻译：0. 1美元的矩阵据说是正常的, 如果它的所有行和列都有相同数量。 我们证明, 对于无限多的整数 $k$, 存在一个平方正正正数 0. 1美元的矩阵, 其二元值为 $k, 其补充的布林级别为 $k ⁇ blobyltilde {Omega} (logkk) 。 等量地, 矩阵中的人可以分割成 $k$ 的组合矩形, 而对于任何零的封面需要的矩形数是 $klobaltilde {Omega} (logkkk) $k 。 这解决了以强烈的形式, Pullman 的问题( liearl Algebra Appl., 1988) 和 Hefner, Henson, Lundgren, 和 ebe (Congrum. Num., 1990) 。 其结果可以被看成一个常规 和 KOS 的直径 的直径直径直径 和直径直径解的直径直径直径数( ) 和直径 的直径 和直径的直径的直径直径直径的直径 。