On a generalization of a result of Kleitman

A classical result of Kleitman determines the maximum number $f(n,s)$ of subsets in a family $\mathcal{F}\subseteq 2^{[n]}$ of sets that do not contain distinct sets $F_1,F_2,\dots ,F_s$ that are pairwise disjoint in the case $n\equiv 0,-1$ (mod $s$). Katona and Nagy determined the maximum size of a family of subsets of an $n$-element set that does not contain $A_1,A_2,\dots ,A_t,B_1,B_2,\dots ,B_t$ with ${\bigcup }_{i=1}^t{A}_i$ and ${\bigcup }_{i=1}^t{B}_i$ being disjoint. In this paper, we consider the problem of finding the maximum number $vex(n,K_{s\times t})$ in a family $\mathcal{F}\subseteq 2^{[n]}$ without sets $F_1^1,\dots ,F_t^1,\dots ,F_1^s,\dots ,F_t^s$ such that $G_j={\bigcup }_{i=1}^t{F}_i^j$ $j=1,2,\dots ,s$ are pairwise disjoint. We determine the asymptotics of $2^n-vex(n,K_{s\times t})$ if $n\equiv -1$ (mod $s$) for all $t$, and if $n\equiv 0$ (mod $s$), $t\ge 3$ and show that in this latter case the asymptotics of the $t=2$ subcase is different from both the $t=1$ and $t\ge 3$ subcases.