Abstract: | The Erdos-Ko-Rado (EKR) theorem is a classical result in extremal set theory. It states that when k < n/2, any family of k-subsets of an n-set X, with the property that any two subsets in the family have nonempty intersection, has size at most binomial(n-1, k-1); equality holds if and only if the family consists of all k-subsets of X containing a fixed point. Here we consider EKR type problems for permutation groups. In particular, we focus on the action of the 2-dimensional projective special linear group PSL(2, q) on the projective line PG(1, q) over the finite field F_q, where q is an odd prime power. A subset S of PSL(2, q) is said to be an intersecting family if for any g1, g2 in S, there exists an element x in PG(1, q) such that g1(x) = g2(x). It is known that the maximum size of an intersecting family in PSL(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. This is a joint work with Ling Long, Rafael Plaza, and Peter Sin. |