Recursive-Line Zarankiewicz Numbers with Four Columns
Abstract
The recursive-line Zarankiewicz number maximizes the number of squares in a structured irreducible sum-of-squares representation encoded by an augmentation of an extremal C4-free bipartite graph. We determine its four-column behavior under the strengthened recursive definition in the manuscript of Löfberg and Qi dated 9 September 2026. Combining AI-assisted discovery with exact certificate verification and finite exclusion computations, we determine eighteen of the nineteen values for 2≤ m≤20 and isolate the only unresolved case to 37≤zr(14,4)≤38. More significantly, we prove the first eventual exact formula in the four-column setting: zr(m,4)=floor(5m+6)/2 (m≥15). The upper bound follows from the classical identity z(m,4)=m+6 and a sharp cell count. For the matching lower bound, we construct a two-row extension chain from an explicit 20×4 seed and derive the odd orders by a fixed deletion. Analytic propagation, together with two independently audited symbolic certificate tables, proves the construction for arbitrary chain length rather than merely for a finite computational range. Thus every extremal configuration has no holes when m is even and exactly one hole when m is odd, and the same exact formula holds for the second-order number z2(m,4).
Related
- A Twelve-Row Seed and a One-Row Extension for Five-Column Recursive-Line Zarankiewicz Numbers
- 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{chen2026recursiveline,
title = {{Recursive-Line Zarankiewicz Numbers with Four Columns}},
author = {Zhiwei Chen and Yannan Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11093},
eprint = {2609.11093},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11093},
}
RIS
TY - UNPB
TI - Recursive-Line Zarankiewicz Numbers with Four Columns
AU - Zhiwei Chen
AU - Yannan Chen
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.11093
UR - https://arxiv.org/abs/2609.11093
AB - The recursive-line Zarankiewicz number maximizes the number of squares in a structured irreducible sum-of-squares representation encoded by an augmentation of an extremal C_4-free bipartite graph. We determine its four-column behavior under the strengthened recursive definition in the manuscript of Löfberg and Qi dated 9 September 2026. Combining AI-assisted discovery with exact certificate verification and finite exclusion computations, we determine eighteen of the nineteen values for 2≤ m≤20 and isolate the only unresolved case to 37≤zr(14,4)≤38. More significantly, we prove the first eventual exact formula in the four-column setting: zr(m,4)=floor(5m+6)/2 (m≥15). The upper bound follows from the classical identity z(m,4)=m+6 and a sharp cell count. For the matching lower bound, we construct a two-row extension chain from an explicit 20×4 seed and derive the odd orders by a fixed deletion. Analytic propagation, together with two independently audited symbolic certificate tables, proves the construction for arbitrary chain length rather than merely for a finite computational range. Thus every extremal configuration has no holes when m is even and exactly one hole when m is odd, and the same exact formula holds for the second-order number z_2(m,4).
ER -