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：一个$0,1$矩阵称为正则矩阵，若其所有行和列均含有相同数量的1。我们证明：对无穷多个整数$k$，存在一个二进制秩为$k$的方形正则$0,1$矩阵，使得其补矩阵的布尔秩为$k^{\widetilde{\Omega}(\log k)}$。等价地，该矩阵中的1可被划分为$k$个组合矩形，而覆盖其零元所需矩形的数量为$k^{\widetilde{\Omega}(\log k)}$。这一结果以强化形式解决了Pullman（Linear Algebra Appl., 1988）提出的问题以及Hefner、Henson、Lundgren和Maybee（Congr. Numer., 1990）的猜想。该结论可视为Balodis、Ben-David、Göös、Jain和Kothari（FOCS, 2021）近期结果的正则类比，其研究动机源于通信复杂度中的团与独立集问题以及图论中（已被证伪的）Alon-Saks-Seymour猜想。作为所构造正则矩阵的应用，我们获得了Alon-Saks-Seymour猜想的正则反例，并证明：对无穷多个整数$k$，存在一个正则图，其双团划分数为$k$，色数为$k^{\widetilde{\Omega}(\log k)}$。