All \(3\)-transitive groups satisfy the strict-Erdős-Ko-Rado property (Q6655687)

Let \(n\) and \(k\) be positive integers such that \(n> 2k\) and let \([n]=\{1,2, \ldots,n\}\). A family \(\mathcal{F}\) of \(k\)-subsets in \([n]\) is intersecting if \(S \cap T \not =\emptyset \) for every \(S,T \in \mathcal{F}\). the family \(\mathcal{F}_{i} = \big \{S \subset [n] \; \big | \; |S|=k, \; i \in S \big \}\), is a canonical example of an intersecting family of size \(\binom{k-1}{n-1}\). The celebrated Erdős-Ko-Rado (EKR) Theorem [\textit{P. Erdős} et al., Q. J. Math., Oxf. II. Ser. 12, 313--320 (1961; Zbl 0100.01902)] states that if \(\mathcal{F}\) is an intersecting family, then \(|F|\leq \binom{k-1}{n-1}\); moreover, the equality holds if and only if \(\mathcal{F}=\mathcal{F}_{i}\), for some \(i \in [n]\).\N\NLet \(G \leq \mathrm{Sym}(n)\) be a transitive permutation group, a subset \(S\) of \(G\) is an intersecting set if, for every \(g_{1}, g_{2} \in S\), there is \(i\in [n]\) such that \(g_{1}(i)=g_{2}(i)\). The stabilizer of a point in \([n]\) and its cosets are intersecting sets of size \(|G|/n\). Such families are called canonical intersecting sets. A result in [\textit{K. Meagher} et al., Eur. J. Comb. 55, 100--118 (2016; Zbl 1333.05306)], states that if \(G\) is a 2-transitive group, then \(|G|/n\) is the size of an intersecting set of maximum size in \(G\). A permutation group, in which every intersecting family of maximum possible size is canonical, is said to satisfy the strict EKR property.\N\NIn the paper under review, the author, using the classification of 3-transitive groups, proves that all 3-transitive groups satisfy the strict-EKR property (a conjecture made in [Meagher et al., loc. cit.]). To do this, the author considers the linear group \(\mathrm{AGL}(n,2)\) with its natural action on the \(2^{n}\) points of the \(n\)-dimensional vector space \(\mathbb{F}_{2}^{n}\) and he shows that such a 3-transitive group satisfies the strict-EKR property. From [\textit{P. J. Cameron}, in: Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland); Cambridge, MA: MIT Press. 611--645 (1995; Zbl 0845.20001)] (Theorem 5.2), it follows that an affine 3-transitive group is either isomorphic to \(\mathrm{AGL}(n,2)\) or \(\mathbb{F}_{2}^{4} \rtimes \mathrm{Alt}(7)\). The author verifies, by computer use, that \(\mathbb{F}_{2}^{4} \rtimes \mathrm{Alt}(7)\) satisfies strict-EKR property. Since the other \(3\)-transitive groups satisfy the strict-EKR property, the author can conclude (Corollary 1.2) that all \(3\)-transitive groups satisfy the strict-Erdős-Ko-Rado property.