A random Hall-Paige Conjecture

Abstract

A transversal in the multiplication table of a group is a set of |G| entries with different rows, columns, & symbols. Which multiplication tables have transversals? An answer was conjectured by Hall & Paige and proved by Wilcox, Evans, and Bray using the classification of finite simple groups. Recently, Eberhard, Manners, and Mrazović found an alternative proof of the Hall-Paige conjecture for sufficiently large groups using ideas from analytic number theory. In this talk, a third proof will be presented using a third set of techniques, this time from probabilistic combinatorics. The new proof is versatile enough to be applicable to a wide range of problems on the boundary of combinatorics and algebra. In particular, conjectures of Snevily, Cichacz, Tannenbaum, Evans, and Keedwall can be solved for sufficiently large groups. Joint work with Alp Müyesser.

Speaker Intro

Alexey Pokrovskiy completed his PhD on the topic of "Graph Powers, Partitions, and other Extremal Problems" under the supervision of Jozef Skokan and Jan van den Heuvel. Since then, he has continued working on extremal combinatorics particularly on the areas of Ramsey theory, Latin squares, and positional games. Prior to joining UCL he was a postdoc at Freie Universitat Berlin and ETH Zurich, and a lecturer at Birkbeck College. Currently he is a lecturer at University College London, and works on problems in-between combinatorics and algebra. In 2019 he received the European Prize in Combinatorics jointly with Richard Montgomery.