On a conjecture of Tokushige for cross-t-intersecting families

2024/10/30 by Huajun Zhang, Biao Wu, Zhang, Huajun +1 · 5 citations
Medicine · Mathematics · #Magnolia and Illicium research #Algebraic Geometry and Number Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2410.22792

Abstract

Two families of sets A and B are called cross-t-intersecting if |A∩ B|≥ t for all A∈ A, B∈ B. An active problem in extremal set theory is to determine the maximum product of sizes of cross-t-intersecting families. This incorporates the classical Erdős--Ko--Rado (EKR) problem. In the present paper, we prove that if A and B are cross-t-intersecting families of \binom [n]k with k≥ t≥ 3 and n≥ (t+1)(k-t+1), then |A||B|≤ \binomn-tk-t²; moreover, if n>(t+1)(k-t+1), then equality holds if and only if A=B is a maximum t-intersecting subfamily of \binom[n]k. This confirms a conjecture of Tokushige for t≥ 3.

Cited by

• Tokushige's conjecture on measures of cross t-intersecting families
• The maximum product of sizes of cross-\(t\)-intersecting families
• The maximum sum of sizes of non-empty cross L-intersecting families
• Cross-intersection theorems for uniform partitions of finite sets
• Extremal cross t-intersecting families under t-covering number constraints for vector spaces

Related

• The maximum product of sizes of cross-\(t\)-intersecting families
• On cross-2-intersecting families
• Exact extremal non-trivial cross-intersecting families
• The maximum product of sizes of cross-t-intersecting uniform families
• Tokushige's conjecture on measures of cross t-intersecting families