The Erdős-Ko-Rado Theorem for Transitive Permutation Groups

Abstract

Given a transitive permutation group \(G \le Sym(\Omega)\), a subset \(F\) of \(G\) is \(\textit {intersecting}\) if any two elements of \(F\) agree on some elements of \(\Omega\). We are interested in the problem of finding the structure of the largest intersecting families of \(G\). This problem is the analogue of the \(\textit {Erdős-Ko-Rado (EKR) Theorem}\) for transitive permutation groups. We say that a transitive group \(G \le Sym(\Omega )\) has the \(\textit{EKR property}\) if any intersecting set of \(G\) has size at most the order of a stabilizer of a point of \(G\). Moreover, \(G\) has the \(\textit {strict-EKR property}\) if the largest intersecting sets in \(G\) are cosets of a stabilizer of a point of \(G\). In this thesis, we use various algebraic techniques to prove EKR-type results for finite transitive groups. In particular, we prove that the action of the symmetric group on the 2-tuples with distinct entries and 2-subsets of \([n]\) have the EKR property, and construct families of transitive groups that are as far away as possible from having the EKR property. Then, we show that any transitive subgroup of \(GL_2(q)\) acting on the non-zero vectors of \(\mathbb F^2_q\) has the EKR property. We also prove that for any odd primes \(p\), the size of the largest intersecting set in a transitive group of degree \(2p\) is at most twice the order of a point stabilizer. In addition, we show that if \(G\) is transitive of degree a product of two odd primes, then \(G\) has the EKR property whenever the socle of \(G\) admits an imprimitive subgroup.