Exact and Asymptotic Values for Weak Limited Augmented Zarankiewicz Numbers in the $m\times 3$ Case

Abstract

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.