# The maximum sum of sizes of cross-intersecting families of subsets of a set

A set of sets is called a family. Two families π and β¬ of sets are said to be cross-intersecting if each member of π intersects each member of β¬. For any two integers n and k with 1 β€ k β€ n, let [n] β€ k denote the family of subsets of [n] = {1, β¦, n} that have at most k elements. We show that if π is a non-empty subfamily of [n] β€ r, β¬ is a non-empty subfamily of [n] β€ s, r β€ s, and π and β¬ are cross-intersecting, then |π| + |β¬| β€ 1 + β_i=1^s (n i - n-r i), and equality holds if π = {[r]} and β¬ is the family of sets in [n] β€ s that intersect [r].

## Authors

• 3 publications
• 10 publications
• ### On the maximum number of distinct intersections in an intersecting family

For n > 2k β₯ 4 we consider intersecting families β± consisting of k-subse...
08/01/2021 β by Peter Frankl, et al. β 0 β

• ### Sharp results concerning disjoint cross-intersecting families

For an \$n\$-element set \$X\$ let \$\binom{X}{k}\$ be the collection of all i...
05/20/2019 β by Peter Frankl, et al. β 0 β

• ### Ranking Sets of Objects: The Complexity of Avoiding Impossibility Results

The problem of lifting a preference order on a set of objects to a prefe...
04/12/2021 β by Jan MalΓ½, et al. β 0 β

• ### Set-Codes with Small Intersections and Small Discrepancies

We are concerned with the problem of designing large families of subsets...
01/16/2019 β by R. Gabrys, et al. β 0 β

• ### Partitioning all k-subsets into r-wise intersecting families

Let r β₯ 2, n and k be integers satisfying k β€r-1/rn. We conjecture that ...
07/27/2021 β by Noga Alon, et al. β 0 β

• ### Structure and properties of large intersecting families

We say that a family of k-subsets of an n-element set is intersecting, i...
10/06/2017 β by Andrey Kupavskii, et al. β 0 β