We give a simple construction of $n\times n$ Boolean matrices with $\Omega(n^{4/3})$ zero entries that are free of $2 \times 2$ all-zero submatrices and have covering number $O(\log^4(n))$. This construction provides an explicit counterexample to a conjecture of Pudl\'{a}k, R\"{o}dl and Savick\'{y} and Research Problems 1.33, 4.9, 11.17 of Jukna [Boolean function complexity]. These conjectures were previously refuted by Katz using a probabilistic construction.
翻译:我们用$\mega(n ⁇ 4/3}) 来简单构建一个纯值n\timen n$bolian 矩阵, 零分数为$\time $2\time 2$ all- 0次矩阵, 覆盖值$O(\log}4(n)) 。 此构造为Pudl\\\ {a}k, R\\"{o}dl和Savick\}{y} 和研究问题1.33, 4.9, 11.17 Jukna [Boolean 函数复杂性] 提供了明确的反推论。 这些推论曾被Katz 使用概率性构造反驳 。