AI 中文总结
本文研究二阶Zarankiewicz数，证明递归线、符号与二阶参数渐近等价，并借助计算机辅助枚举确定若干宽度下的精确值。
AI 中文摘要
递归线和符号Zarankiewicz数在满足两个充分不可约性准则的前提下，最大化最大$C_4$-自由基的增广中的方格数。计数包括每个基单元一个方格和每对未使用单元一个方格。我们将这些参数与使用不可约性本身的二阶数进行比较。每个最大的$m\times n$基都允许一个递归线增广，其方格数至少为$mn/2-C\max(m,n)$，其中$C$为绝对常数。将此界与已知固定宽度族的两列扩展相结合，我们证明所有三个参数渐近等价，且当较大维度趋于无穷时一致成立。对于单个展示，传递图表明一旦符号闭包识别出每一对选定单元，符号准则就等价于不可约性。我们确定了每个六列矩形的二阶数，并给出了宽度为七、九和十一时所有三个数的最终精确公式。我们还证明了在$8\times7$处符号与递归线的相等性，并且结合早期值，当较短边至多为六时（除可能在$14\times4$处外）也成立。精确值结果是计算机辅助的，使用穷举枚举、经过检验的命题反驳和符号证书，并带有适用于任意长度的已证明提升论证。
英文摘要
The recursive-line and signed Zarankiewicz numbers maximize the number of squares in augmentations of a maximum $C_4$-free base, subject to two sufficient irreducibility criteria. The count includes one square per base cell and one per selected pair of unused cells. We compare these parameters with the second-order number, which uses irreducibility itself. Every maximum $m\times n$ base admits a recursive-line augmentation with at least $mn/2-C\max(m,n)$ squares, for an absolute constant $C$. Combining this bound with a two-column extension of known fixed-width families, we show that all three parameters are asymptotically equivalent, uniformly as the larger dimension tends to infinity. For individual displays, a transfer graph shows that once the signed closure identifies every selected pair, the signed criterion is equivalent to irreducibility. We determine the second-order number for every six-column rectangle and give eventual exact formulas for all three numbers at widths seven, nine and eleven. We also prove signed and recursive-line equality at $8\times7$ and, together with earlier values, whenever the shorter side is at most six, except possibly at $14\times4$. The exact-value results are computer-assisted, using exhaustive enumeration, checked propositional refutations and symbolic certificates with a proved lifting argument for arbitrary lengths.
Comments30 pages. Supporting evidence: https://doi.org/10.5281/zenodo.23124764
