Zarankiewicz数z(m,n;3,3)的五个改进下界
Five improved lower bounds for Zarankiewicz numbers z(m,n;3,3)
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中文总结 AI 辅助
本文给出Zarankiewicz数z(m,n;3,3)的五个改进下界，通过kicked-greedy局部搜索生成的无K_{3,3}的0/1矩阵等完成验证，改进幅度明确且经多方独立核验。
中文摘要 AI 辅助
本文记录了s=t=3时Zarankiewicz数的五个改进下界：z(13,19;3,3)≥118，z(14,19;3,3)≥126，z(16,18;3,3)≥136，z(14,20;3,3)≥126，z(16,19;3,3)≥136。前三个下界由明确的无K_{3,3}的0/1矩阵提供证明，后两个通过单调填充由第二个和第三个下界推导得出。与arXiv:2605.01120图2中汇总的下界（分别为114、121、130、125、132）相比，改进幅度分别为+4、+5、+6、+1、+4。这三个矩阵由发现系统自主运行的kicked-greedy局部搜索生成，经多个独立实现对所有3×3行/列三元组进行穷举检查后精确验证；arXiv:2605.01120的作者也在2026年7月使用自身验证器独立重新验证了这些矩阵。所有三个见证矩阵完整刊载于附录A，并作为机器可读的辅助文件及独立无依赖的验证器随本笔记提供。
英文摘要
We record five improved lower bounds for Zarankiewicz numbers with s = t = 3: z(13,19;3,3) >= 118, z(14,19;3,3) >= 126, z(16,18;3,3) >= 136, z(14,20;3,3) >= 126, z(16,19;3,3) >= 136. The first three are certified by explicit K_{3,3}-free 0/1 matrices; the last two follow from the second and third by monotone padding. Compared with the lower bounds compiled in Figure 2 of arXiv:2605.01120, namely 114, 121, 130, 125 and 132, the improvements are +4, +5, +6, +1 and +4 respectively. The three matrices were produced by a kicked-greedy local search operated autonomously by a discovery system and were verified exactly, by exhaustive inspection of every 3x3 row/column triple, in several mutually independent implementations; they were also re-verified independently by the authors of arXiv:2605.01120 using their own verifier in July 2026. All three witnesses are printed in full in Appendix A and accompany this note as machine-readable ancillary files together with standalone, dependency-free verifiers.
