Non-uniform skew versions of Bollobás' Theorem

Abstract

Let $A_1,\dots ,A_m$ and $B_1,\dots ,B_m$ be subsets of $[n]$ and let $t$ be a non-negative integer with the following property: $|A_i\cap B_i|\le t$ for each $i$ and $|A_i\cap B_j|>t$ whenever $i<j$. Then $m\le 2^{n-t}$. Our proof uses Lovász' tensor product method. We prove the following skew version of Bollobás' Theorem. Let $A_1,\dots ,A_m$ and $B_1,\dots ,B_m$ be finite sets of $[n]$ satisfying the conditions $A_i\cap B_i=\varnothing$ for each $i$ and $A_i\cap B_j\ne \varnothing$ for each $i<j$. Then $$\sum _{i=1}^m\frac{1}{\left(\genfrac{}{}{0ex}{}{|A_i|+|B_i|}{|A_i|}\right)}\le n+1.$$ Both upper bounds are sharp.