The EPPA–Ramsey expansion conjecture

A class of finite structures has EPPA if every structure in the class has a finite EPPA-witness, meaning a finite structure into which it embeds such that every partial automorphism extends to an automorphism. An expansion is precompact if each structure has only finitely many expansions, up to isomorphism, in the expanded language. A class has the Ramsey property when every finite structure and every number of colors admit a larger structure whose copies of the given structure are monochromatic under every coloring of those copies.

EPPA–Ramsey expansion conjecture. Every class with EPPA has a precompact Ramsey expansion.

This conjecture proposes a general connection between the extension property for partial automorphisms and structural Ramsey theory. The paper presents it as a motivating conjecture for the classification of EPPA classes; examples such as finite groups and finite skew-symmetric structures are mentioned as cases where the existence of a precompact Ramsey expansion remains open.