### Matchings in hypergraphs defined by groups

- Series
- Graph Theory Seminar
- Time
- Tuesday, February 14, 2023 - 03:45 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Alp Muyesser – University College London – alp.muyesser.21@ucl.ac.uk

When can we find perfect matchings in hypergraphs whose vertices represent group elements and whose edges represent solutions to systems of linear equations? A prototypical problem of this type is the Hall-Paige conjecture, which asks for a characterisation of the groups whose multiplication table (viewed as a Latin square) contains a transversal. Other problems expressible in this language include the toroidal n-queens problem, Graham-Sloane harmonious tree-labelling conjecture, Ringel's sequenceability conjecture, Snevily's subsquare conjecture, Tannenbaum's zero-sum conjecture, and many others. All of these problems have a similar flavour, yet until recently they have been approached in completely different ways, using algebraic tools ranging from the combinatorial Nullstellensatz to Fourier analysis. In this talk we discuss a unified approach to attack these problems, using tools from probabilistic combinatorics. In particular, we will see that a suitably randomised version of the Hall-Paige conjecture can be used as a black-box to settle many old problems in the area for sufficiently large groups. Joint work with Alexey Pokrosvkiy