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- W4296404629 abstract "For an $n$-element set $X$ let $binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $mathcal A$ and $mathcal B$ are called cross-intersecting if $Acap B neq emptyset$ holds for all $Ainmathcal A$, $Binmathcal B$. Let $f(n,k)$ denote the maximum of $min{|mathcal A|, |mathcal B|}$ where the maximum is taken over all pairs of {em disjoint}, cross-intersecting families $mathcal A, mathcal Bsubsetbinom{[n]}{k}$. Let $c=log_2e$. We prove that $f(n,k)=leftlfloorfrac12binom{n-1}{k-1}rightrfloor$ essentially iff $n>ck^2$ (cf. Theorem~1.4 for the exact statement). Let $f^*(n,k)$ denote the same maximum under the additional restriction that the intersection of all members of both $mathcal A$ and $mathcal B$ are empty. For $kge5$ and $nge k^3$ we show that $f^*(n,k)=leftlfloorfrac12left(binom{n-1}{k-1}-binom{n-2k}{k-1}right)rightrfloor+1$ and the restriction on $n$ is essentially sharp (cf. Theorem~5.4)." @default.
- W4296404629 created "2022-09-20" @default.
- W4296404629 creator A5048570599 @default.
- W4296404629 creator A5056866010 @default.
- W4296404629 date "2019-05-20" @default.
- W4296404629 modified "2023-09-29" @default.
- W4296404629 title "Sharp results concerning disjoint cross-intersecting families" @default.
- W4296404629 doi "https://doi.org/10.48550/arxiv.1905.08123" @default.
- W4296404629 hasPublicationYear "2019" @default.
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