A Proof of Füredi's Conjecture

We prove Füredi's conjecture on strong Bollobás $t$-systems. For every nonnegative integer $t$, a family of $m\ge2$ pairs of finite sets $(A_i,B_i)$ satisfying $|A_i\cap B_i|\le t$ and $|A_i\cap B_j|>t$ for all $i\ne j$ satisfies \[ \sum_i\binom{|A_i|+|B_i|-2t}{|A_i|-t}^{-1}\le1. \] We first prove a weighted inequality for subspace pairs satisfying the symmetric cross-intersection condition over any field. It follows from a local inequality for graded exterior ideals, proved by induction with projections and colon ideals.