# Pure Maths Colloquium: Wolfram Bentz

This talk is part of the Pure Maths Colloquium at the University of St Andrews. Check out our upcoming talks at https://theran.lt/pure-colloquium/.

 Where: Lecture Theatre D When: Jan 31 2019 @ 16.00 Video: Not recorded
 Speaker: Wolfram Bentz University of Hull Title: Regularity and the existential transversal property

Let $$G$$ be a permutation group of degree $$n$$ on the domain $$\Omega$$, and $$k$$ a positive integer with $$k\le n$$. We say that $$G$$ has the $$k$$-existential property, or $$k$$-et, if there exists a $$k$$-subset $$A$$ whose orbit under $$G$$ contains transversals for all $$k$$-partitions $$\mathcal{P}$$ of $$\Omega$$.

This property is a substantial weakening of the $$k$$-universal transversal property, or $$k$$-ut, investigated by my coauthors, which required this condition to hold for all $$k$$-subsets $$A$$ of the domain $$\Omega$$.

Both of these conditions relate to the regularity of transformation semigroups. In the earlier work, the regularity of $$\langle G,t\rangle$$ for any map $$t$$ of rank $$k$$ (with $$k < n/2$$) was shown to be equivalent to the $$k$$-ut property. The question investigated in our new result is if there is a $$k$$-subset $$A$$ of the domain such that $$\langle G, t\rangle$$ is regular for all maps $$t$$ with image $$A$$. This turns out to be much more delicate: the $$k$$-et property (with $$A$$ as witnessing set) is a necessary, but not sufficient condition.

In this talk we give a nearly complete characterizations of both $$k$$-et and its corresponding regularity condition in the case that $$4\le k\le n/2$$.

This is joint work with João Araújo (Universidade Nova) and Peter J. Cameron (St Andrews).