Monomorphic structures (structures with only one kind of n-element substructures, for each n) were introduced and studied by R. Fraïssé as natural generalizations of chains ( \(=\) linear orders). This notion was later generalized by Pouzet and Thierý to structures admitting a finite monomorphic decomposition. In this paper we characterize countable structures admitting a finite monomorphic decomposition which have finite big Ramsey degrees. The necessary prerequisite for that is the characterization of monomorphic structures with finite big Ramsey degrees. Interestingly, both characterizations require deep structural properties of chains. Fraïssé’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 of chains. We find this last result particularly intriguing because big Ramsey degrees are known to exhibit irregular behavior when it comes to general product statements. As a spin-off of the product Ramsey theorem, we provide an alternative proof of Hubička’s result that the generic partial order has finite big Ramsey degrees.
