Exact big Ramsey degrees for finitely constrained binary free amalgamation classes

TL;DR

This paper precisely determines the big Ramsey degrees for certain finitely constrained binary free amalgamation classes, refining previous bounds and establishing the existence of strong big Ramsey structures for their Fraïssé limits.

Contribution

It provides an exact characterization of big Ramsey degrees for classes defined by finitely many forbidden substructures, advancing understanding of their combinatorial and topological properties.

Findings

Exact big Ramsey degrees characterized for these classes

Fraïssé limits admit strong big Ramsey structures

Automorphism groups have metrizable universal completion flows

Abstract

We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization, we show that the Fra\"iss\'e limit of each such class admits a strong big Ramsey structure, implying that the automorphism group of the Fra\"iss\'e limit has a metrizable universal completion flow.