2022/12/19 by Adam Kabela, Michal Polák, Kabela, Adam +3 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2212.09279
openalex publication_date 2022/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We let F be a finite family of sets closed under taking unions and ∅ \not ∈ F, and call an element abundant if it belongs to more than half of the sets of F. In this notation, the classical Frankl's conjecture (1979) asserts that F has an abundant element. As possible strengthenings, Poonen (1992) conjectured that if F has precisely one abundant element, then this element belongs to each set of F, and Cui and Hu (2019) investigated whether F has at least k abundant elements if a smallest set of F is of size at least k. Cui and Hu conjectured that this holds for k = 2 and asked whether this also holds for the cases k = 3 and k > (n)/(2) where n is the size of the largest set of F. We show that F has at least k abundant elements if k ≥ n - 3, and that F has at least k - 1 abundant elements if k = n - 4, and we construct a union-closed family with precisely k - 1 abundant elements for every k and n satisfying n - 4 ≥ k ≥ 3 and n ≥ 9 (and for k = 3 and n = 8). We also note that F always has at least min \ n, 2k - n + 1 \ abundant elements. On the other hand, we construct a union-closed family with precisely two abundant elements for every k and n satisfying n ≥ max \ 3, 5k-4 \. Lastly, we show that Cui and Hu's conjecture for k = 2 stands between Frankl's conjecture and Poonen's conjecture.