# 07-21【Qing Xiang 】 Zoom-吴文俊数学重点实验室组合图论系列讲座之152

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 {1,2,...,n}, with the property that any two subsets in the family have nonempty intersection, has size at most {n-1 \choose k-1}; equality holds if and only if the family consists of all k-subsets of {1,2,...,n} containing a fixed element.

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 Fq, 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 x^g1 = x^g2. 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 must be cosets of point stabilizers for all odd prime powers q>3. This talk is based on joint work with Ling Long, Rafael Plaza, and Peter Sin.