两个有限边界切片上的精确Zarankiewicz值
Exact Zarankiewicz Values On Two Finite Frontier Slices
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中文总结 AI 辅助
本文针对二分图的Zarankiewicz数，通过计算机辅助组合证明，得到了多个有限边界切片的精确Zarankiewicz值，部分值以区间形式给出，相关结论可通过标准Python工具复现。
中文摘要 AI 辅助
Zarankiewicz数Z(m,n,s,t)是指顶点数分别为m和n的二分图中不含Ks,t子图的最大边数。本文针对两个有限切片及修正后的相邻边界给出了基于组合证明的计算机辅助证明，得到如下精确值：Z(12,n,3,3)=6n（18≤n≤22），Z(13,22,3,3)=137，Z(13,18,3,3)=116，Z(14,18,3,3)=124，Z(15,18,3,3)=132，Z(14,17,3,3)=118，Z(15,17,3,3)=126，Z(16,17,3,3)介于132到133之间。关键的新上界是精确的12×18和13×18证书包，其轨道证书排除了所有达到下一个边数的假设矩阵；删除引理和显式见证关闭了四个相邻单元，而16×17条目被报告为区间，因为仅验证了其132边的下见证和已发表的133上界。此外，13×22的证明通过简化为83个度分布，合理分离其中77个，再通过标记行同余、叶子枚举、模Gram检验和精确Farkas证书消除剩余6个，从而排除了138的可能值。所有已验证的结论均可通过标准库Python及精确整数/有理算术复现，仅在发现证书时使用浮点优化。
英文摘要
The Zarankiewicz number Z(m,n,s,t) is the maximum number of edges in a bipartite graph with parts of orders m and n containing no copy of Ks,t. We give one combined, certificate-based computer-assisted proof for two finite slices and a corrected neighboring frontier: Z(12,n,3,3) = 6n (18 <= n <= 22), Z(13,22,3,3) = 137, Z(13, 18, 3, 3) = 116, Z(14, 18, 3, 3) = 124, Z(15,18,3,3) = 132, Z(14, 17, 3, 3) = 118, Z(15, 17, 3, 3) = 126, 132 <= Z(16,17,3,3) <= 133. The load-bearing new upper bounds are the exact 12 x 18 and 13 x 18 certificate packages. Their orbit certificates exclude every hypothetical matrix at the next edge count. Deletion lemmas and explicit witnesses close four neighboring cells, while the 16 x 17 entry is deliberately reported as an interval because only its 132-edge lower witness and the published 133 upper bound are certified here. Separately, the 13 x 22 proof excludes 138 ones by reducing to 83 degree profiles, rationally separating 77 of them, and eliminating the remaining six by marked-row congruences, leave enumeration, modular Gram tests, and exact Farkas certificates. All accepted claims are replayed by standard-library Python and exact integer/rational arithmetic; floating-point optimization is used only to discover certificates.
