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- W4299656554 abstract "A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Families $mathcal{A}_1, mathcal{A}_2, dots, mathcal{A}_k$ of sets are 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, for a 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 any union $mathcal{H}$ 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 be avoided). Our primary result asserts that there are only two possible optimal configurations for the sum. A special case was conjectured by Kamat in 2011. We also prove generalizations, whereby $mathcal{A}_1, mathcal{A}_2, dots, mathcal{A}_k$ are not necessarily contained in the same union of levels. Various ErdH{o}s-Ko-Rado-type results follow. The sum problem for a level of a power set was solved for $t=1$ by Hilton in 1977, and for any $t$ by Wang and Zhang in 2011." @default.
- W4299656554 created "2022-10-02" @default.
- W4299656554 creator A5038574319 @default.
- W4299656554 date "2018-05-14" @default.
- W4299656554 modified "2023-09-28" @default.
- W4299656554 title "Cross-intersecting subfamilies of levels of hereditary families" @default.
- W4299656554 doi "https://doi.org/10.48550/arxiv.1805.05241" @default.
- W4299656554 hasPublicationYear "2018" @default.
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