A family of subsets of [n] is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl–Kupavskii and Balogh–Das–Liu–Sharifzadeh–Tran showed that for n≥2k+cklnk, almost all k-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for n≥2k+100lnk. Our proof uses, among others, the graph container method and the Das–Tran removal lemma.
- Extremal combinatorics
- Graph container method
- Intersecting family
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics