Intersecting families of sets are usually trivial for n≥ 2k+3

2026/08/01 by Jiabao Yang
Mathematics · #math.CO #msc:05D05

paper · pdf

20 pages

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

A family of subsets of [n] is called intersecting if it contains no pair of disjoint sets. It is called trivial if all its members contain a common element. Frankl and Kupavskii, and independently Balogh, Das, Liu, Sharifzadeh, and Tran, proved that there is a constant c>0 such that, whenever n ≥ 2k+2+c√(kln k), almost all k-uniform intersecting families are trivial. Balogh, Garcia, Li, and Wagner later improved this range to n ≥ 2k+100ln k. In this paper, we prove that the same conclusion holds for every n≥ 2k+3. This verifies the conjectured conclusion of Balogh, Garcia, Li, and Wagner throughout this range.

Citations

• Improved bounds on the maximum diversity of intersecting families
• Intersecting families of sets are typically trivial
• Diversity of uniform intersecting families
• Counting intersecting and pairs of cross-intersecting families
• Stability versions of Erdős-Ko-Rado type theorems, via isoperimetry
• Intersecting families of discrete structures are typically trivial
• INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS