On Füredi’s conjecture

Abstract

We confirmed the following special case of Füredi’s conjecture: Let \(t\) be a non-negative integer. Let \( \mathcal{ P}=\{(A_i,B_i)\}_{1\leq i\leq m}\) be a set-pair family satisfying \(|A_i \cap B_i|\leq t\) for \(1\leq i \leq m\) and \(|A_i\cap B_j|>t\) for all \(1\leq i\neq j \leq m\). Define \(a_i:=|A_i|\) and \(b_i:=|B_i|\) for each \(i\). Assume that there exists a positive integer \(N\) such that \(a_i+b_i=N\) for each \(i\). Then

$$\sum_{i=1}^m \frac{1}{{a_i+b_i-2t \choose a_i-t}}\leq 1.$$