Intersecting $k$-Uniform Families Containing all the $k$-Subsets of a Given Set

Wei-Tian Li, Bor-Liang Chen, Kuo-Ching Huang, Ko-Wei Lih


Let $m, n$, and $k$ be integers satisfying $0 < k \leq n < 2k \leq m$. A family of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\binom{[n]}{k} \subseteq \mathcal{F} \subseteq \binom{[m]}{k}$ and any pair of members of $\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of Erdős-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large.


intersecting family; cross-intersecting family; Erdős-Ko-Rado;Milner-Hilton; Kneser graph

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