Ramsey multiplicity of patterns in abelian groups
Burr and Rosta conjectured that given any fixed (small) graph H, a random 2-colouring of the edges of the complete graph K_n contains (asymptotically) the minimum number of monochromatic copies of H. This conjecture was disproved by Sidorenko in 1989, who showed that it is false when H is a triangle with a pendant edge. A weaker conjecture by Erdos was disproved almost simultaneously by Thomason. Despite a number of other special cases having been resolved since, the general classification problem remains wide open. We explore an arithmetic analogue of this question, which turns out to have implications for the original graph theoretic problem.
(This is joint work with Alex Saad.)