The Zarankiewicz number z(33;2) equals 195
Year: 2026
Type: preprint
Abstract: The Zarankiewicz number z(n;2) is the largest number of ones in an n x n zero-one matrix with no 2 x 2 all-ones submatrix, equivalently the largest number of edges of a C4-free bipartite graph with n vertices in each part. Collins, Riasanovsky, Wallace and Radziszowski (2016) tabulated its values through n=31, and their bound z(32;2) <= 190 gives 189 <= z(33;2) <= 202. We prove that z(33;2) = 195; together with z(32;2) = 189, this determines z(n;2) for every n <= 33. The lower bound is a matrix obtained from the projective plane of order 5. For the upper bound we exclude matrices with 197 and with 196 ones; both parts use the value z(32;2) = 189. For 197 ones, the structure of the unique row and column of degree 5 reduces the problem to seven cases, which are excluded by hand and by exhaustive enumeration of the remaining free part followed by satisfiability computations. For 196 ones, degree lemmas and exact linear screens reduce the pairs of row and column degree profiles to 48 ordered pairs. These are excluded by hand arguments, by an analysis of the Gram matrix of the rows through the graph of disjoint pairs of rows, including the Matrix-Tree theorem and a positive-semidefiniteness bound, and by satisfiability computations. Every unsatisfiability verdict is backed by a proof checked by an independent proof checker. This record contains the manuscript (PDF) and the supplementary archive, split into five parts (join them as described in JOIN.txt): the code, the extremal matrix and its 720 variants, the enumeration and sieve outputs, the exact certificates, the formulas or their generators with digests, the deposited LRAT proofs and the case ledger, with a SHA-256 manifest and instructions for reproducing every computation. MSC2020: 05C35 (primary); 05C38, 05C85, 68R10 (secondary)
Author: Yan, Yancun
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222065899