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A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ commonelements. Families $\mathcal{A}_1, \mathcal{A}_2, \dots, \mathcal{A}_k$ of setsare cross-$t$-intersecting if, for every $i$ and $j$ in $\{1, 2, \dots, k\}$with $i \neq j$, each set in $\mathcal{A}_i$ $t$-intersects each set in$\mathcal{A}_j$. An active problem in extremal set theory is to determine, fora given finite family $\mathcal{F}$, the structure of $k$cross-$t$-intersecting subfamilies whose sum or product of sizes is maximum.For a family $\mathcal{H}$, the $r$-th level $\mathcal{H}^{(r)}$ of$\mathcal{H}$ is the family of all sets in $\mathcal{H}$ of size $r$, and, for$s \leq r$, $\mathcal{H}^{(s)}$ is called a $(\leq r)$-level of $\mathcal{H}$.We solve the problem for any union $\mathcal{F}$ of $(\leq r)$-levels of anyunion $\mathcal{H}$ of power sets of sets of size at least a certain integer$n_0$, where $n_0$ is independent of $\mathcal{H}$ and $k$ but depends on $r$and $t$ (dependence on $r$ is inevitable, but dependence on $t$ can beavoided). Our primary result asserts that there are only two possible optimalconfigurations for the sum. A special case was conjectured by Kamat in 2011. Wealso prove generalizations, whereby $\mathcal{A}_1, \mathcal{A}_2, \dots,\mathcal{A}_k$ are not necessarily contained in the same union of levels.Various Erd\H{o}s-Ko-Rado-type results follow. The sum problem for a level of apower set was solved for $t=1$ by Hilton in 1977, and for any $t$ by Wang andZhang in 2011.

Cross-intersecting subfamilies of levels of hereditary families