A Twelve-Row Seed and a One-Row Extension for Five-Column Recursive-Line Zarankiewicz Numbers
Abstract
We prove that the five-column recursive-line Zarankiewicz number satisfies zRL(m,5)=3m+5 for every integer m≥ 12. The proof is based on an explicit 12×5 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 (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 z2(m,5)=zRL(m,5)=3m+5 throughout this range.
Related
- Recursive-Line Zarankiewicz Numbers with Four Columns
- Overtake and the recursive-line equality for second-order Zarankiewicz numbers
- Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?
- New Bounds for Limited Zarankiewicz Numbers from K5t Blocks
- Density and separation for augmented Zarankiewicz numbers
BibTeX
@misc{xi2026twelverow,
title = {{A Twelve-Row Seed and a One-Row Extension for Five-Column Recursive-Line Zarankiewicz Numbers}},
author = {Min Xi and Jingya Chang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33191},
eprint = {2609.33191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33191},
}
RIS
TY - UNPB
TI - A Twelve-Row Seed and a One-Row Extension for Five-Column Recursive-Line Zarankiewicz Numbers
AU - Min Xi
AU - Jingya Chang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33191
UR - https://arxiv.org/abs/2609.33191
AB - We prove that the five-column recursive-line Zarankiewicz number satisfies z_RL(m,5)=3m+5 for every integer m≥ 12. The proof is based on an explicit 12×5 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 (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.
ER -