On a conjecture of Tokushige for cross-$t$-intersecting families
Huajun Zhang, Biao Wu

TL;DR
This paper proves a conjecture in extremal set theory, establishing the maximum product size of cross-$t$-intersecting families of sets under certain conditions, extending the classical Erdős–Ko–Rado problem.
Contribution
It confirms Tokushige's conjecture for $t ext{-}ge 3$, providing a tight bound on the product of sizes of cross-$t$-intersecting families.
Findings
Maximum product of sizes is ${inom{n-t}{k-t}}^2$ under specified conditions.
Equality holds if and only if the families are equal and maximum $t$-intersecting.
Results extend classical intersection theorems to cross-$t$-intersecting families.
Abstract
Two families of sets and are called cross--intersecting if for all , . An active problem in extremal set theory is to determine the maximum product of sizes of cross--intersecting families. This incorporates the classical Erd\H{o}s--Ko--Rado (EKR) problem. In the present paper, we prove that if and are cross--intersecting families of with and , then ; moreover, if , then equality holds if and only if is a maximum -intersecting subfamily of . This confirms a conjecture of Tokushige for .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMagnolia and Illicium research · Algebraic Geometry and Number Theory · Meromorphic and Entire Functions
