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Size, diversity, minimum degree, sturdiness, dömdödömBalázs Patkós, 2026, original scientific article
Abstract: For a family F of sets and a disjoint pair A, B we let F(A, overline(B)) = F ∈ F: A ⊆ F, B ∩ F = ∅}.The (p,q)-dömdödöm of a family F ⊆ 2[n] is β(p, q)(F)=min {|F(A, overline(B))|:|A|=p,|B|=q, A ∩ B = ∅, A, B ⊆ [n]}. This definition encompasses size, diversity, minimum degree, and sturdiness as special cases. We investigate the maximum possible value β(p, q)(n,k) of β(p, q)(F) over all k-uniform intersecting families F ⊂ 2[n]. We determine the order of magnitude of β(p, q)(n,k) for all fixed p,q,k. We relate the asymptotics of β(p, q)(n,k) to the constant value of β(0, q)(n,q+1) and establish β(p, 1)(n,k)=\binom{n-3-p}{k-2-p} and β(p, 2)(n,k)= 2\binom{n-5}{k-3-p} -\binom{n-7}{k-5-p} if n is large enough.
Keywords: intersecting families, diversity, minimum degree
Published in RUP: 11.08.2026; Views: 213; Downloads: 2
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