The maximum size of the partial ground set of skew Bollob\'{a}s systems

Abstract

A skew Bollob\'{a}s system $\mathcal{P}=\{(A_i,B_i):1\leq i\leq m\}$ is a collection of pairs of disjoint subsets of $[n]$ such that $A_i\cap B_j\ne\emptyset$ for any $1\leq i<j\leq m$. Denote by $S_1(a, b)$ or $S_2(a, b)$ the maximum size of $\bigcup_{i=1}^m A_i$ or $\bigcup_{i=1}^m B_i$, respectively, over all possible skew Bollob\'{a}s systems $\mathcal{P}=\{(A_i,B_i):1\leq i\leq m\}$ satisfying $|A_i| \leq a$ and $|B_i| \leq b$ for all $i \in [m]$. It is shown that for any non-negative integers $a$ and $b$, $S_1(a,b)=\binom{a+b+1}{a}-1$ and $S_2(a,b)=\binom{a+b+1}{a+1}-1$.