Geodesic
Intersecting \(k\)-uniform families containing all the \(k\)-subsets of a given set
Wei-Tian Li1 ; Bor-Liang Chen2 ; Kuo-Ching Huang3 ; Ko-Wei Lih4
1National Chung Hsing University
2National Taichung University of Science and Technology
3Providence University
4Academia Sinica
The electronic journal of combinatorics, Tome 20 (2013) no. 3
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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.
DOI : 10.37236/3213
Classification : 05D05
Mots-clés : intersecting family, cross-intersecting family, Erdős-Ko-Rado theorem, Milner-Hilton theorem, Kneser graph

Wei-Tian Li  1   ; Bor-Liang Chen  2   ; Kuo-Ching Huang  3   ; Ko-Wei Lih  4

1 National Chung Hsing University
2 National Taichung University of Science and Technology
3 Providence University
4 Academia Sinica 
@article{10_37236_3213,
     author = {Wei-Tian Li and Bor-Liang Chen and Kuo-Ching Huang and Ko-Wei Lih},
     title = {Intersecting \(k\)-uniform families containing all the \(k\)-subsets of a given set},
     journal = {The electronic journal of combinatorics},
     year = {2013},
     volume = {20},
     number = {3},
     doi = {10.37236/3213},
     zbl = {1295.05250},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/3213/}
}
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Wei-Tian Li; Bor-Liang Chen; Kuo-Ching Huang; Ko-Wei Lih. Intersecting \(k\)-uniform families containing all the \(k\)-subsets of a given set. The electronic journal of combinatorics, Tome 20 (2013) no. 3. doi: 10.37236/3213

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