m×3情形下弱受限增广Zarankiewicz数的新下界
Exact and Asymptotic Values for Weak Limited Augmented Zarankiewicz Numbers in the $m\times 3$ Case
AI总结：
该研究确定了m=5至9时的弱受限增广Zarankiewicz数，验证了对应关系，通过构造与搜索得到上下界，支持相关猜想并推导了BSR数的下界。
AI中文摘要：
我们确定了当$m=5,6,7,8,9$时精确的弱受限增广Zarankiewicz数$z_{wL}(m,3)$，建立了$z_{wL}(5,3)=10$、$z_{wL}(6,3)=12$、$z_{wL}(7,3)=14$、$z_{wL}(8,3)=16$、$z_{wL}(9,3)=18$，因此对于所有满足$5\te m\te 9$的$m$，都有$z_{wL}(m,3)=2m$。下界由显式的弱容许构造给出，而$m=7,8,9$时的匹配上界则通过对无$C_4$的规范基础图进行精确有限搜索得到。结合已知的小情形，这为所有$m\te 5$时$z_{wL}(m,3)=2m$的猜想提供了有力支撑。由于BSR$(m,n)\te z_{wL}(m,n)$，这些构造意味着对于$m=5,6,7,8,9$，有${\rm BSR}(m,3)\te 2m$。
英文摘要：
We determine the exact weak limited augmented Zarankiewicz numbers $z_{wL}(m,3)$ for all $m\ge 3$: \[ z_{wL}(m,3)= \begin{cases} m+3+\left\lceil \dfrac{m}{2}\right\rceil+1, & 9\le m\le 15,\\[2mm] m+3+\left\lfloor \dfrac{2m-4}{3}\right\rfloor, & m\ge 16, \end{cases} \] with $z_{wL}(3,3)=6$, $z_{wL}(4,3)=8$, and $z_{wL}(m,3)=2m$ for $5\le m\le 9$. In particular, \[ \lim_{m\to\infty} \frac{z_{wL}(m,3)}{m} = \frac{5}{3}. \] The proof is fully analytic, relying on a uniform base classification, two constructive lower-bound families (staircase and $5m/3$), and a sharp upper-bound argument based on a peeling lemma and the analysis of two W2-sensitive boundary cases. Numerical MILP computations were used only as proof-mining tools to identify the structural lemmas; the final theorem is unconditional. We also extend the known range of the original limited numbers $z_L(m,3)$ through $m=13$, where the gap to $z_{wL}(m,3)$ is only 2 or 3.
