Füredi 猜想的证明

中文摘要 AI 辅助

我们证明了 Füredi 关于强 Bollobás $t$-系统的猜想。对于每个非负整数 $t$，一个由 $m\ge2$ 对有限集 $(A_i,B_i)$ 组成的族，若满足 $|A_i\cap B_i|\le t$ 且对所有 $i\ne j$ 有 $|A_i\cap B_j|>t$，则满足 \\[ \sum_i\binom{|A_i|+|B_i|-2t}{|A_i|-t}^{-1}\le1. \\] 我们首先证明一个关于子空间对的加权不等式，该子空间对在任意域上满足对称交叉相交条件。它由分次外部理想的局部不等式得出，该不等式通过投影和余理想进行归纳证明。

英文摘要

We prove a strengthening of Füredi's conjecture on strong Bollobás $t$-systems. For nonnegative integers $t$ and $s$, a family of $m\ge2$ pairs of finite sets $(A_i,B_i)$ satisfying $|A_i\cap B_i|\le t$ and $|A_i\cap B_j|>t+s$ for all $i\ne j$ satisfies \[ \sum_{i=1}^{m} \frac{\binom{|B_i|-t+s}{s}} {\binom{|A_i|+|B_i|-2t}{|A_i|-t}}\le1. \] When $s=0$, this is precisely Füredi's conjectured weighted inequality. The proof is based on a local growth inequality for graded exterior ideals, a basis-exchange construction, and a common-section reduction. The same method gives the corresponding inequality for subspaces over arbitrary fields. We also determine a best possible upper bound for the Füredi weight sum, construct equality examples with an arbitrary prescribed number of pairs, and derive further inequalities from two-sided basis exchanges.

发表机构

• School Of Mathematics, Hunan University（湖南大学数学学院）

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