Füredi's Conjecture and a Sharp Strengthening

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}{