A note on the maximum size of the ground set of skew Bollobás systems

Abstract

A skew Bollobás system $D=\left\{\right(A_i^{\left(1\right)},\dots ,A_i^{\left(d\right)}):1\le i\le m\}$ is a collection of d pairwise disjoint subsets of $\left[n\right]$ such that for any $1\le i<j\le m$, there exist $1\le p<q\le d$ with $A_i^{\left(p\right)}\cap A_j^{\left(q\right)}\ne \varnothing$. Denote by $n_{\text{s}kew}(a_1,\dots ,a_d)$ the maximum size of the ground set ${\bigcup }_{i=1}^m{\bigcup }_{r=1}^d{A}_i^{\left(r\right)}$ of a skew Bollobás system $D$ such that $\left|A_i^{\left(r\right)}\right|\le a_r$ for $i\in \left[m\right]$ and $r\in \left[d\right]$. We show that for any positive integers $a_1,\dots ,a_d$,$n_{\text{s}kew}(a_1,\dots ,a_d)=\sum _{j=1}^{{\sum }_{r=1}^d{a}_r-1}\sum _{\stackrel{{\lambda }_1+\cdots +{\lambda }_d=j}{0\le {\lambda }_r\le a_r,\forall r\in \left[d\right]}}\left(\begin{array}{c}\sum _{r=1}^d{a}_r-j\\ a_1-{\lambda }_1,\dots ,a_d-{\lambda }_d\end{array}\right)+1.$ In particular for $d=2$, we have$n_{\text{s}kew}(a_1,a_2)=\left(\begin{array}{c}{a}_1+a_2+2\\ a_1+1\end{array}\right)-\left(\begin{array}{c}{a}_1+a_2\\ a_1\end{array}\right)-1$ for any non-negative integers $a_1$ and $a_2$. This corrects a typo in the upper bound on $n_{skew}(a_1,a_2)$ given by Nagy and Patkós (2015) [7].