Wednesday, December 23 2020
15:00 - 16:00

IMSc Webinar

EKR problems for permutation groups

Pantangi Venkata Raghu Tej

SUSTech, Shenzen

Zoom Meeting ID: 813 6569 0574 Password: Littlewood.

Erdős-Ko-Rado (EKR) theorem is a classical result in extremal set theory. It characterizes the largest collection of pair-wise $k$-subsets of an $n$-set. An active line of research is to generalize this result to different objects. Let $G$ be a transitive permutation group on a set $\Omega$. A subset $\mathcal{F}\subset G$ is said to be an intersecting set if any two permutations $g,h\in \mathcal{F}$ agree on a point $\omega \in \Omega$, that is, $\omega^{g}=\omega^{h}$. Cosets of point stabilizers are natural examples of intersecting sets. An intersecting set is said to be a maximum intersecting set if it is of the maximum possible size. In view of the classical EKR theorem, it is of interest to characterize maximum intersecting sets. A group is said to satisfy the EKR property if for every intersecting set $\mathcal{F}$, we have $|\mathcal{F}|\leq|G_{\omega}|$, that is, cosets of point stabilizers are maximum intersecting sets. It is known that if $G$ is either Frobenius or $2$-transitive, it satisfies the EKR property. In this talk, we will see that general transitive permutation groups are quite far from satisfying the EKR property. In particular, we show that even in the case of primitive groups, there is no absolute constant $c$ such that $|\mathcal{F}|\leqslant c|G_\omega|$. This is joint work with Cai Heng Li and Shu Jiao Song.

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