About Füredi's conjecture

Abstract

Let $t$ be a non-negative integer and $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ be a set-pair family satisfying $|A_i\cap B_i|\le t$ for $1\le i\le m$. $\mbox{$\cal P$}$ is called strong Bollobás $t$-system, if $|A_i\cap B_j|>t$ for all $1\le i\ne j\le m$. Füredi conjectured the following nice generalization of Bollobás' Theorem: Let $t$ be a non-negative integer. Let $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ be a strong Bollobás $t$-system. Then $$\sum _{i=1}^m\frac{1}{\left(\genfrac{}{}{0ex}{}{|A_i|+|B_i|-2t}{|A_i|-t}\right)}\le 1.$$ We confirmed the following special case of Füredi's conjecture along with some more results of similar flavor. Let $t$ be a non-negative integer. Let $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ denote a strong Bollobás $t$-system. 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 $$\sum _{i=1}^m\frac{1}{\left(\genfrac{}{}{0ex}{}{a_i+b_i-2t}{a_i-t}\right)}\le 1.$$