2020/10/01 by Behajaina, Angelot, Maleki, Roghayeh, Rasoamanana, Aina Toky +1
#05C35 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.00229
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\.