New Lower Bounds for Weak Limited Augmented Zarankiewicz Numbers in the $m\times 3$ Case

Liqun Qi; Yannan Chen

New Lower Bounds for Weak Limited Augmented Zarankiewicz Numbers in the $m\times 3$ Case

Optimization and Control 2026-08-06 v1

Authors: Liqun Qi , Yannan Chen

Abstract

We determine the exact weak limited augmented Zarankiewicz numbers $z_{wL}(m,3)$ for $m=5,6,7,8,9$ . We establish $z_{wL}(5,3)=10,\quad z_{wL}(6,3)=12,\quad z_{wL}(7,3)=14,\quad z_{wL}(8,3)=16,\quad z_{wL}(9,3)=18.$ Thus $z_{wL}(m,3)=2m$ for all $5\le m\le 9$ . The lower bounds are given by explicit weak admissible constructions, while the matching upper bounds for $m=7,8,9$ are obtained by exact finite searches over canonical $C_4$ -free base graphs. Together with the known small cases, this strongly supports the conjecture that $z_{wL}(m,3)=2m \quad \text{for all } m\ge 5.$ Since BSR $(m,n)\ge z_{wL}(m,n)$ , these constructions imply ${\rm BSR}(m,3)\ge 2m \quad \text{for } m=5,6,7,8,9.$

Keywords

Zarankiewicz numbers weak limited augmented construction C4-free base graphs lower bounds bipartite Ramsey numbers

Cite

@article{arxiv.2608.06050,
  title  = {New Lower Bounds for Weak Limited Augmented Zarankiewicz Numbers in the $m\times 3$ Case},
  author = {Liqun Qi and Yannan Chen},
  journal= {arXiv preprint arXiv:2608.06050},
  year   = {2026}
}