Non-uniform skew versions of Bollob\'as' Theorem

Abstract

Let $A_1, \ldots ,A_m$ and $B_1, \ldots ,B_m$ be subsets of $[n]$ and let $t$ be a non-negative integer with the following property: $|A_i \cap B_i|\leq t$ for each $i$ and $|A_i\cap B_j|>t$ whenever $i< j$. Then $m\leq 2^{n-t}$. Our proof uses Lov\'asz' tensor product method. We prove the following skew version of Bollob\'as' Theorem. Let $A_1, \ldots ,A_m$ and $B_1, \ldots ,B_m$ be finite sets of $[n]$ satisfying the conditions $A_i \cap B_i =\emptyset$ for each $i$ and $A_i\cap B_j\ne \emptyset$ for each $i< j$. Then $$ \sum_{i=1}^m \frac{1}{{|A_i|+|B_i| \choose |A_i|}}\leq n+1. $$ Both upper bounds are sharp.