The maximum size of the partial ground set of skew Bollobás systems

A skew Bollobás system $\mathcal{P}=\{(A_i,B_i):1\le i\le m\}$ is a collection of pairs of disjoint subsets of $[n]$ such that $A_i\cap B_j\mathrm{\ne }\mathrm{\varnothing }$ for any $1\le i<j\le m$. 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 Bollobás systems $\mathcal{P}=\{(A_i,B_i):1\le i\le m\}$ satisfying $\mathrm{\mid }{A}_i\mathrm{\mid }\le a$ and $\mathrm{\mid }{B}_i\mathrm{\mid }\le b$ for all $i\in [m]$. It is shown that for any non-negative integers $a$ and $b$, $S_1(a,b)=\left(\genfrac{}{}{0px}{}{a+b+1}{a}\right)-1$ and $S_2(a,b)=\left(\genfrac{}{}{0px}{}{a+b+1}{a+1}\right)-1$.