Papers › $3$-setwise intersecting families of the symmetric group
$3$-setwise intersecting families of the symmetric group
Angelot Behajaina, Roghayeh Maleki, Aina Toky Rasoamanana, A. Sarobidy Razafimahatratra
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Given two positive integers n≥3 and t≤n, the permutations σ,π∈Sym(n) are t-setwise intersecting if they agree (setwise) on a t-subset of {1,2,…,n}. A family ℱ ⊂Sym(n) is t-setwise intersecting if any two permutations of ℱ are t-setwise intersecting. Ellis [Journal of Combinatorial Theory, Series A, 119(4), 825--849, 2012] conjectured that if t≤n and ℱ ⊂Sym(n) is a t-setwise intersecting family, then |ℱ|≤t!(n-t)! and equality holds only if ℱ is a coset of a setwise stablizer of a t-subset of {1,2,…,n}. In this paper, we prove that if n≥11 and ℱ is $3$-setwise intersecting, then |ℱ|≤6(n-3)!. Moreover, we prove that the characteristic vector of a $3$-setwise intersecting family of maximum size lies in the sum of the eigenspaces induced by the permutation module of Sym(n) acting on the $3$-subsets of {1,2,…,n}.
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