Finite big Ramsey degrees for generalized pseudotrees

Determine whether, for every finite set of branching orders P contained in {3, 4, …}, the rooted pseudotree Ψ_P has finite big Ramsey degrees.

Background

The paper discusses generalized rooted pseudotrees Ψ_P arising from the ramification points of generalized Ważewski dendrites, where P specifies the permitted finite branching orders. Kwiatkowska’s result that the age of the corresponding finite rooted-tree class has a precompact Ramsey expansion motivates asking whether the associated countable ultrahomogeneous structure necessarily has finite big Ramsey degrees.

The authors explicitly pose this question for every finite P but restrict their investigation to the special case P = {3}, the two-branching pseudotree Ψ. The paper proves that finite chains in this special pseudotree have finite big Ramsey degrees, while antichains of size at least two have infinite big Ramsey degree; it therefore does not resolve the question for all finite branching sets P.