Ramsey Cayley graphs for symmetric groups

Determine whether the family of symmetric groups admits Ramsey Cayley graphs.

Background

Symmetric groups simultaneously contain large elementary abelian subgroups, large cyclic subgroups, and composition factors of large rank. The paper identifies them as a barrier to all of the methods used to prove its Ramsey-Cayley results.

Resolving this question would establish Alon’s conjecture for a fundamental non-abelian family and would likely require ideas substantially different from the approximate-group and subgroup-avoidance techniques developed in the paper.