Fra\"{i}ss\'{e}'s Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition

TL;DR

This paper characterizes when countable structures with finite monomorphic decomposition have finite big Ramsey degrees, linking structural properties, Fra"{i}ssé's Conjecture, and a product Ramsey theorem.

Contribution

It provides a new characterization of structures with finite big Ramsey degrees using monomorphic parts and deep structural properties of chains.

Findings

Countable monomorphic structures have finite big Ramsey degrees iff they are chainable with finite big Ramsey degrees.

Finite big Ramsey degrees are characterized by the structure of monomorphic parts in minimal decompositions.

A product Ramsey theorem for big Ramsey degrees is established, addressing complex combinatorial behavior.

Abstract

In this paper we show that a countable structure admitting a finite monomorphic decomposition has finite big Ramsey degrees if and only if so does every monomorphic part in its minimal monomorphic decomposition. The necessary prerequisite for this result is the characterization of monomorphic structures with finite big Ramsey degrees: a countable monomorphic structure has finite big Ramsey degrees if and only if it is chainable by a chain with finite big Ramsey degrees. Interestingly, both characterizations require deep structural properties of chains. Fra\"{i}ss\'{e}'s Conjecture (actually, its positive resolution due to Laver) is instrumental in the characterization of monomorphic structures with finite big Ramsey degrees, while the analysis of big Ramsey combinatorics of structures admitting a finite monomorphic decomposition requires a product Ramsey theorem for big Ramsey degrees. We find this last result particularly intriguing because big Ramsey degrees misbehave notoriously when it comes to general product statements. In the addendum, we combine a recent result by Oudrar and Pouzet with our analysis of finite big Ramsey degrees for structures admitting finite monomorphic decomposition to characterize the existence of finite Big Ramsey degrees for all countable relational structures whose language has a linear order and age has polynomial growth.