Infinite-dimensional Ramsey theory for homogeneous structures with SDAP$^+$

TL;DR

This paper establishes that certain homogeneous structures satisfying SDAP$^+$ have subspaces of the Baire space where all Borel sets are Ramsey, leading to new results on big Ramsey degrees and topological Ramsey spaces.

Contribution

It proves a Ramsey theorem for homogeneous structures with SDAP$^+$, including new topological Ramsey spaces and exact big Ramsey degrees, answering longstanding questions.

Findings

All Borel sets in certain subspaces are Ramsey.

Exact big Ramsey degrees are recovered for these structures.

Constructs topological Ramsey spaces satisfying Ellentuck-type theorems.

Abstract

We prove that for any homogeneous structure $\mathbf{K}$ in a language with finitely many relation symbols of arity at most two satisfying SDAP$^+$ (or LSDAP$^+$), there are spaces of subcopies of $\mathbf{K}$, forming subspaces of the Baire space, in which all Borel sets are Ramsey. Structures satisfying SDAP$^+$ include the rationals, the Rado graph and more generally, unrestricted structures, and generic $k$-partite graphs, the latter three types with or without an additional dense linear order. As a corollary of the main theorem, we obtain an analogue of the Nash-Williams Theorem which recovers exact big Ramsey degrees for these structures, answering a question raised by Todorcevic at the 2019 Luminy Workshop on Set Theory. Moreover, for the rationals and similar homogeneous structures our methods produce topological Ramsey spaces, thus satisfying analogues of the Ellentuck theorem.