【摘要】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.

【报告人简介】向青，1995获美国 Ohio State University 博士学位, 现为美国特拉华大学 (University of Delaware) 教授。主要研究方向为组合设计、有限几何、编码和加法组合。现为国际组合数学界权威期刊《The Electronic Journal of Combinatorics》主编，同时担任SCI期刊《Designs, Codes and Cryptography》, 《Journal of Combinatorial Designs》的编委。曾获得国际组合数学及其应用协会颁发的杰出青年学术成就奖—Kirkman Medal。在国际组合数学界最高级别杂志《J. Combin. Theory Ser. A》,《J. Combin. Theory Ser. B》, 《Combinatorica》,以及《Trans. Amer. Math. Soc.》,《IEEE Trans. Inform. Theory》等重要国际期刊上发表学术论文80余篇。主持完成美国国家自然科学基金、美国国家安全局等科研项目10余项。曾在国际学术会议上作大会报告或特邀报告50余次。