Characterization of intersecting families of maximum size in PSL(2,q)

Abstract

We consider the action of the 2-dimensional projective special linear group P SL(2, q) on the projective line P G(1, q) over the finite field F q , where q is an odd prime power. A subset S of P SL(2, q) is said to be an intersecting family if for any g 1 , g 2 ∈ S, there exists an element x ∈ P G(1, q) such that x g1 = x g2 . It is known that the maximum size of an intersecting family in P SL(2, q) is q(q − 1)/2. We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers q > 3.

Supercongruences Occurred to Rigid Hypergeometric Type Calabi–Yau Threefolds

“…A different application of character sums H p (α, β; 1) to problems in discrete mathematics can be found in [35].…”

The Erdős-Ko-Rado theorem for the derangement graph of the projective general linear group acting on the projective space

“…Suppose that α, β, γ, δ are as in Cases 6,10,11,13,14,15,17. The fact that in all of these cases C (α,β),(γ,δ) equals v is a miracle in our opinion and we do not fully understand the reason of this behaviour. We first consider Case 6.…”

“…Recently, Long, Plaza, Sin and Xiang [14] have proved the Erdős-Ko-Rado theorem for the derangement graph of PSL 2 (q) in its action of on the projective line: this result was conjectured in [15]. In their work, these authors have developed some new ideas and they posed a beautiful conjecture [14,Section 6] concerning the stability of extremal intersecting families in PSL 2 (q), in the same spirit as the results proved by Ellis [4] on the stability of the extremal intersecting families in Sym(n).…”

“…Recently, Long, Plaza, Sin and Xiang [14] have proved the Erdős-Ko-Rado theorem for the derangement graph of PSL 2 (q) in its action of on the projective line: this result was conjectured in [15]. In their work, these authors have developed some new ideas and they posed a beautiful conjecture [14,Section 6] concerning the stability of extremal intersecting families in PSL 2 (q), in the same spirit as the results proved by Ellis [4] on the stability of the extremal intersecting families in Sym(n). Now that [15,Conjecture 2] is proved, we remark the relevance and importance of the work in [14] for a possible strengthening of Theorem 1.1, and we encourage investigations on the stability of extremal intersecting families for general projective linear groups.…”

An Erdős-Ko-Rado theorem for the group $$\hbox {PSU}(3, q)$$ PSU ( 3 , q )

“…Although all 2-transitive groups have the ERK-property, it is not true that all 2-transitive groups have the strict-EKR property. It has been shown that Sym(n), Alt(n), PGL(2, q), PGL(3, q), PSL(2, q) [14] and the Mathieu groups all have the strict-EKR property by first showing that they have the EKR-module property [1,2,10,15,16,14]. It is shown in [17] that the Suzuki groups, Ree Groups, Higman-Sims, Symplectic groups also have the EKR-module property.…”