Füredi's Conjecture and a Sharp Strengthening

Abstract

We prove a strengthening of Füredi's conjecture on strong Bollobás $t$-systems. For nonnegative integers $t$ and $s$, a family of $m\ge 2$ 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.