The Erdős-Ko-Rado property for some 2-transitive groups

An Erdös--Ko--Rado Theorem for the Derangement Graph of ${PGL}_3(q)$ Acting on the Projective Plane

“…They both further showed that the only independent sets meeting this bound are the cosets of the stabilizer of a point. The same result was also proved in [8] using the character theory of Sym(n).Recently there have been many papers questioning if the natural extension of the Erdős-Ko-Rado theorem holds for specific permutation groups G (see [2,9,11,13]). This means asking if the largest independent sets in the derangement graph Γ G are the cosets in G of the stabilizer of a point.…”

“…The only property of PGL 3 (q) that is used in the proof of Proposition 3.2 is the 2-transitivity. In fact, this result for any 2-transitive group is shown in [2].…”

“…Recently there have been many papers questioning if the natural extension of the Erdős-Ko-Rado theorem holds for specific permutation groups G (see [2,9,11,13]). This means asking if the largest independent sets in the derangement graph Γ G are the cosets in G of the stabilizer of a point.…”

An Erdős–Ko–Rado theorem for finite 2-transitive groups

“…Recently there have been many papers proving that the natural extension of the Erdős-Ko-Rado theorem holds for specific permutation groups G (see [1,13,26,28,29,33]) and there are also two papers, [2] and [3], that consider when the natural extension of the Erdős-Ko-Rado theorem holds for transitive and 2transitive groups. Again, this means asking if the largest intersecting sets in G are the cosets in G of the stabiliser of a point.…”

“…is a subspace of CG with dimension χ 0 (1) 2 + ψ(1) 2 = 1 + (|Ω| − 1) 2 . Now, [28,Proposition 3.2] and also [2] show that V = χ S | S coset of the stabiliser of a point has also dimension 1 + (|Ω| − 1) 2 . Since W ≤ V , we get W = V and hence Conjecture 1.3 holds for G.…”