Almost eventowns

Wednesday, 9 August 2023

Abstract

A town, in which people have formed clubs, is called an eventown if any of its two, not necessarily distinct, clubs share an even number of members. It is called an almost eventown, if out of every three of its clubs, at least two share an even number of members.

It follows from a particular case of a conjecture of Erdős, that the maximum number of clubs in an almost eventown of a given size is about twice that of an eventown of the same size (the exact value of the latter had been previously conjectured by Erdős and settled independently by Berlekamp (1969) and Graver (1975))

I will present a proof of the conjecture based on linear algebra and character sums and will also discuss a number of analogues, generalisations and related open problems

This talk is based on joint work with Giorgis Petridis