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 \(\textrm{vex}(n,K_{s\times t})\) in a family \({{\mathcal {F}}}\subseteq 2^{[n]}\) without sets \(F^1_1,\dots ,F^1_t,\dots ,F^s_1,\dots ,F^s_t\) such that \(G_j=\bigcup _{i=1}^tF^j_i\) \(j=1,2,\dots ,s\) are pairwise disjoint. We determine the asymptotics of \(2^n-\textrm{vex}(n,K_{s\times t})\) 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.
