Exact Zarankiewicz Values On Two Finite Frontier Slices

Abstract

The Zarankiewicz number Z(m,n,s,t) is the maximum number of edges in a bipartite graph with parts of orders m and n containing no copy of Ks,t. We give one combined, certificate-based computer-assisted proof for two finite slices and a corrected neighboring frontier: Z(12,n,3,3) = 6n (18 <= n <= 22), Z(13,22,3,3) = 137, Z(13, 18, 3, 3) = 116, Z(14, 18, 3, 3) = 124, Z(15,18,3,3) = 132, Z(14, 17, 3, 3) = 118, Z(15, 17, 3, 3) = 126, 132 <= Z(16,17,3,3) <= 133. The load-bearing new upper bounds are the exact 12 x 18 and 13 x 18 certificate packages. Their orbit certificates exclude every hypothetical matrix at the next edge count. Deletion lemmas and explicit witnesses close four neighboring cells, while the 16 x 17 entry is deliberately reported as an interval because only its 132-edge lower witness and the published 133 upper bound are certified here. Separately, the 13 x 22 proof excludes 138 ones by reducing to 83 degree profiles, rationally separating 77 of them, and eliminating the remaining six by marked-row congruences, leave enumeration, modular Gram tests, and exact Farkas certificates. All accepted claims are replayed by standard-library Python and exact integer/rational arithmetic; floating-point optimization is used only to discover certificates.

Disclosure

“EXACT ZARANKIEWICZ VALUES ON TWO FINITE FRONTIER SLICES 11 Acknowledgements OpenAI GPT 5.6 Sol High was used for assistance with exploratory reasoning, implemen- tation, verification, manuscript integration, and packaging. The author is solely responsible for the mathematical claims and final manuscript. No heuristic solver stat”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T05:01:08.208275Z

Structural counts