2018/04/30 by Hao Huang, Huang, Hao · 2 citations
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1804.11269
openalex publication_date 2018/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families A, B ⊂ \binom[n]k, they must satisfy min\|A|, |B|\ ≤ (1)/(2) \binomn-1k-1? We give an affirmative answer for n ≥ 2k2, and construct families showing that this range is essentially the best one could hope for, up to a constant factor. The second problem is a conjecture of Frankl. It states that for n ≥ 3k, the maximum diversity of an intersecting family F ⊂ \binom[n]k is equal to \binomn-3k-2. We are able to find a construction beating the conjectured bound for n slightly larger than 3k, which also disproves a conjecture of Kupavskii.