On Füredi’s conjecture

Abstract

AbstractWe confirmed the following special case of Füredi’s conjecture: Let $$t$$ t be a non-negative integer. Let $$ \mathcal{ P}=\{(A_i,B_i)\}_{1\leq i\leq m}$$ P = { ( A i , B i ) } 1 ≤ i ≤ m be a set-pair family satisfying $$|A_i \cap B_i|\leq t$$ | A i ∩ B i | ≤ t for $$1\leq i \leq m$$ 1 ≤ i ≤ m and $$|A_i\cap B_j|>t$$ | A i ∩ B j | > t for all $$1\leq i\neq j \leq m$$ 1 ≤ i ≠ j ≤ m . Define $$a_i:=|A_i|$$ a i : = | A i | and $$b_i:=|B_i|$$ b i : = | B i | for each $$i$$ i . Assume that there exists a positive integer $$N$$ N such that $$a_i+b_i=N$$ a i + b i = N for each $$i$$ i . Then $$\sum_{i=1}^m \frac{1}{{a_i+b_i-2t \choose a_i-t}}\leq 1.$$ ∑ i = 1 m 1 a i + b i - 2 t a i - t ≤ 1 .