Abstract

A typical problem in extremal set theory is to determine how small or how large a parameter of a system of sets can be. The Erd\H{o}s—Ko—Rado~Theorem is a classical result in this field. A variant of the Erd\H{o}s—Ko—Rado problem is to determine the maximum sum or the maximum product of sizes of k cross-t-intersecting subfamilies $\mathcal{A}_1, \mathcal{A}_2, \dots, \mathcal{A}_k$ of a given family $\mathcal{F}$ of sets, where by `cross-$t$-intersecting’ we mean that, for every $i$ and $j$ with $i \neq j$, each set in $\mathcal{A}_i$ intersects each set in $\mathcal{A}_j$ in at least $t$ elements. This natural problem has recently attracted much attention. Solutions have been obtained for various important families, such as power sets, levels of power sets, hereditary families, families of permutations, and families of integer sequences. The talk will provide an outline of these results. It will focus mostly on the product problem for the family of subsets of $\{1, 2, \dots, n\}$ that have at most $r$ elements.