An improved upper bound for the Zarankiewicz number z(43;2) <= 294 via structural rigidity, including hexary packing number pa(5;6) = 31

Abstract

The Zarankiewicz number z(n;2) is the largest number of ones in an n x n zero-one matrix with no all-one 2x2 submatrix. At n=43, the Kovari-Sos-Turan bound, sharpened by Reiman, gives z(43;2) <= 301 with no rounding slack; equality would force a projective plane of order six, which Tarry ruled out in 1900. That fact alone says nothing about how far below 301 the truth lies. We prove z(43;2) <= 294. The argument is local and geometric. Writing D = 301 - E for the deficiency of a configuration with E ones, a crossing count paired with its dual forces every line to at most eight points and every point to at most eight lines whenever D <= 6. A deficiency count then produces a clean point of degree seven, collinear with all others, unless the configuration has both a line of eight points and a point of degree eight. The seven lines through a clean point partition the remaining 42 points into groups that every other line meets at most once, so the number of (line, missed group) incidences equals D. At D=6 this forces group sizes 6,6,6,6,6,6,6 or 7,6,6,6,6,6,5; the second forces a TD(4,6), and the first yields 32 blocks of a partial TD(5,6), whose refutation needs an unknown packing number. We determine it. pa(5;6) and pa(6;6) are listed as open in the Handbook of Combinatorial Designs at 30 <= N <= 34; a class holds at most six words, so at most one full class gives at most 6 + 5*5 = 31, and what was missing was refuting the two-full-class case, done here by exhaustive search. Hence pa(5;6) = pa(6;6) = 31. With no clean point, the configuration is rigid enough to complete to a projective plane of order six. With a known 290-one configuration, this gives 290 <= z(43;2) <= 294. We also derive z(43;2) <= 299 via Totten's classification, and rule out configurations with 290+ ones from deleting rows/columns of PG(2,7).