发表机构
School of Mathematics and Statistics, Guangdong University of Foreign Studies; School of Mathematics and Statistics, Guangdong University of Technology（广东外语外贸大学数学与统计学院; 广东工业大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过显式12×5种子配置与递归单行扩展，证明五列递归线Zarankiewicz数在m≥12时等于3m+5，并利用距离缩减矩形引理统一验证强化条件，结合参数层级得出z_2(m,5)同值。
AI 中文摘要
我们证明五列递归线Zarankiewicz数满足$z_{RL}(m,5)=3m+5$对所有整数$m\ge 12$成立。证明基于一个显式的$12\times5$种子配置，并结合递归单行扩展方案。该种子达到五列单元格界且不含未占用单元格。种子中的两对选定对被打断并替换为平行路径，确保每插入一行使增强边总数增加三，同时保持所需的简单配置。主要挑战在于统一验证所有扩展长度下的强化递归线条件${\rm (RW3+)}$。为此，我们发展了一个距离缩减矩形引理，该引理将认证的内积关系沿扩展路径传递，将远距离标签间的关系归结为较小路径距离上的关系。这将任意长扩展的验证限制为有限集合的种子和接口证书。所得构造成功满足对识别，保持不同选定边类，并认证不同边代表的正交性。此外，将该构造的下界与参数层级结合，得到在此范围内$z_2(m,5)=z_{RL}(m,5)=3m+5$。
英文摘要
We prove that the five-column recursive-line Zarankiewicz number satisfies $z_{RL}(m,5)=3m+5$ for every integer $m\ge 12$. The proof is based on an explicit $12\times5$ seed configuration combined with a recursive one-row extension scheme. The seed attains the five-column cell bound and contains no unoccupied cells. Two selected pairs in the seed are opened and replaced by parallel paths, ensuring that each inserted row increases the total number of augmented edges by three while preserving the required simple configuration. The main challenge lies in verifying the strengthened recursive-line condition ${\rm (RW3+)}$ uniformly across all extension lengths. To this end, we develop a distance-reducing rectangle lemma that transfers certified inner-product relations along extension paths, reducing relations between distant labels to those at smaller path distances. This confines the verification for arbitrarily long extensions to a finite collection of seed and interface certificates. The resulting construction successfully satisfies pair identification, preserves distinct selected-edge classes, and certifies the orthogonality of distinct edge representatives. Furthermore, combining this constructed lower bound with the parameter hierarchy yields $z_2(m,5)=z_{RL}(m,5)=3m+5$ throughout this range.
