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- W2798522807 abstract "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 $mathcal{A}, mathcal{B} subset binom{[n]}{k}$, they must satisfy $min{|mathcal{A}|, |mathcal{B}|} le frac{1}{2} binom{n-1}{k-1}$? We give an affirmative answer for $n ge 2k^2$, 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 $n ge 3k$, the maximum diversity of an intersecting family $mathcal{F} subset binom{[n]}{k}$ is equal to $binom{n-3}{k-2}$. We are able to find a construction beating the conjectured bound for $n$ slightly larger than $3k$, which also disproves a conjecture of Kupavskii." @default.
- W2798522807 created "2018-05-07" @default.
- W2798522807 creator A5041400284 @default.
- W2798522807 date "2018-04-30" @default.
- W2798522807 modified "2023-09-24" @default.
- W2798522807 title "Two extremal problems on intersecting families" @default.
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- W2798522807 doi "https://doi.org/10.48550/arxiv.1804.11269" @default.
- W2798522807 hasPublicationYear "2018" @default.
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