The Zarankiewicz number z(44;2) is 302 or 303
Year: 2026
Type: preprint
Abstract: Preprint and supplementary data for the paper "The Zarankiewicz number z(44;2) is 302 or 303". The Zarankiewicz number z(n;2) is the maximum number of ones in an n×n 0/1 matrix with no 2×2 submatrix of ones. For n = q^2+q+2 and a prime power q, a projective plane of order q gives z(n;2) ≥ (q+1)(q^2+q+1)+3. The first order without a projective plane is q = 6, where n = 44. We prove z(44;2) ≤ 303, improving the previous bound 310; together with the lower bound 302, obtained from an elliptic semiplane of Baker by deleting an incident point and line, this gives z(44;2) ∈ {302, 303}. The proof combines counting arguments for near-extremal C4-free matrices with the non-existence of a projective plane, of two orthogonal Latin squares and of an affine plane of order six, and with finite enumerations whose code and outputs are deposited here. MSC2020: 05C35 (primary); 05B25, 05B30 (secondary). Files: Yan_z44.pdf: the preprint. Yan_z44_Supplementary_archive.zip: programs, outputs and the 44×44 witness matrix with 302 ones; README.txt inside gives the command and the expected result for every computation; run_all.sh reruns all verification computations (about three minutes); SHA256SUMS.txt lists every file. Zip: 36,130 bytes, md5 70ade79c6ef8ebbe8e65dbab4444340d, sha256 ca6cd3355309afab914aac104debb156f758d9f7a8f6229d22154a20a30aba7a (33 members). PDF: md5 2fa1939ff08baeb193066915a2ecbfa7.
Author: Yan, Yancun
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222018318
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