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arXiv · 2609.33191

A Twelve-Row Seed and a One-Row Extension for Five-Column Recursive-Line Zarankiewicz Numbers

Min Xi·Jingya Chang
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Abstract

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.

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Min Xi, Jingya Chang. 2026-09-27. A Twelve-Row Seed and a One-Row Extension for Five-Column Recursive-Line Zarankiewicz Numbers. https://arxiv.org/abs/2609.33191

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