AI 中文总结
本文改进了Zarankiewicz数z(43;2)的上界，证明其不超过299，还在PG(2,7)中构造出284边的无4-圈二分图，得到该数的区间为[284,299]。
AI 中文摘要
Zarankiewicz数z(43;2)是两个部分大小均为43的无4-圈二分图的最大边数。Reiman界给出z(43;2)≤301，仅当存在6阶射影平面的关联图时等号成立；但该平面不存在，故z(43;2)≤300。本文证明z(43;2)≤299。该论证为初等方法，未使用计算机搜索：构型的“剩余部分”计数恒等式表明，假设的300边图每侧仅允许27种度分布中的一种，其中仅4种组合局部相容。3种组合迫使两个顶点共享6个邻点；第4种组合迫使横截设计TD(6,6)，即6阶的4个相互正交拉丁方，与Tarry定理矛盾。本文还在PG(2,7)内给出显式的284边构造，故284≤z(43;2)≤299。
英文摘要
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).
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