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.
We consider the action of the $2$-dimensional projective general linear group $PGL(2,q)$ on the projective line $PG(1,q)$. A subset $S$ of $PGL(2,q)$ is said to be an intersecting family if for every $g_1,g_2 \in S$, there exists $\alpha \in PG(1,q)$ such that $\alpha^{g_1}= \alpha^{g_2}$. It was proved by Meagher and Spiga that the intersecting families of maximum size in $PGL(2,q)$ are precisely the cosets of point stabilizers. We prove that if an intersecting family $S \subset PGL(2,q)$ has size close to the maximum then it must be "close" in structure to a coset of a point stabilizer. This phenomenon is known as stability. We use this stability result proved here to show that if the size of $S$ is close enough to the maximum then $S$ must be contained in a coset of a point stabilizer.
Abstract. Let n ≥ r ≥ s ≥ 0 be integers and F a family of r-subsets of [n]. Let W F r,s be the higher inclusion matrix of the subsets in F vs. the s-subsets of [n]. When F consists of all r-subsets of [n], we shall simply write W r,s in place of W F r,s . In this paper we prove that the rank of the higher inclusion matrix W r,s over an arbitrary field K is resilient. That is, if the size of F is "close" to n r then rank K (W F r,s ) = rank K (W r,s ), where K is an arbitrary field. Furthermore, we prove that the rank (over a field K) of the higher inclusion matrix of r-subspaces vs. s-subspaces of an n-dimensional vector space over F q is also resilient if char(K) is coprime to q.
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