Big Ramsey degrees using parameter spaces

Abstract

We show that the universal homogeneous partial order has finite big Ramsey degrees and discuss several corollaries. Our proof uses parameter spaces and the Carlson-Simpson theorem rather than (a strengthening of) the Halpern-Läuchli theorem and the Milliken tree theorem, which are the primary tools used to give bounds on big Ramsey degrees elsewhere (originating from work of Laver and Milliken). This new technique has many additional applications. To demonstrate this, we show that the homogeneous universal triangle-free graph has finite big Ramsey degrees, thus giving a short proof of a recent result of Dobrinen.

Exact big Ramsey degrees via coding trees

“…We refer to the paper [10] for comprehensive background on historical and recent results on big Ramsey degrees. This paper extends and concludes work in [8], [9], [11], [12], [13], [14], [17], [18], [25], [28], and [32] for free amalgamation classes with relations of arity at most two and finitely many forbidden irreducible substructures.…”

Ramsey theory of homogeneous structures: current trends and open problems

“…Hubička recently developed a new method to handle forbidden substructures utilizing topological Ramsey spaces of parameter words due to Carlson and Simpson [7]. In [39], he applied his method to prove that the homogeneous partial order and Urysohn S-metric spaces (where S is a set of non-negative reals with 0 ∈ S satisfying the 4-values condition) have finite big Ramsey degrees. He also showed that this method is quite broad and can be applied to yield a short proof of finite big Ramsey degrees in G 3 .…”

“…He also showed that this method is quite broad and can be applied to yield a short proof of finite big Ramsey degrees in G 3 . Beginning with the upper bounds in [39], the exact big Ramsey degrees of the generic partial order have been characterized in [2] by Balko, Chodounský, Hubička, Konečný, Vena, Zucker, and the author. Also utilizing techniques from [39], Balko, Chodounský, Hubička, Konečný, Nešetřil, and Vena in [4] have found a condition which guarantees finite big Ramsey degrees for binary relational homogeneous structures with strong amalgamation.…”

“…Beginning with the upper bounds in [39], the exact big Ramsey degrees of the generic partial order have been characterized in [2] by Balko, Chodounský, Hubička, Konečný, Vena, Zucker, and the author. Also utilizing techniques from [39], Balko, Chodounský, Hubička, Konečný, Nešetřil, and Vena in [4] have found a condition which guarantees finite big Ramsey degrees for binary relational homogeneous structures with strong amalgamation. Examples of structures satisfying this condition include the S-Urysohn space for finite distance sets S, Λ-ultrametric spaces for a finite distributive lattice, and metric spaces associated to metrically homogeneous graphs of a finite diameter from Cherlin's list with no Henson constraints.…”

“…We conclude this section by mentioning the exact big Ramsey degrees in the generic partial order in [2]. This result begins with the upper bounds proved by Hubička in [39] and then proceeds by taking a diagonal antichain D representing the generic partial order with additional structure of interesting levels built into D. A level ℓ of D is interesting if there are exactly two nodes, say s, t, in that level so that ( * ) for exactly one relation ρ ∈ {<, >, ⊥}, given any s ′ , t ′ ∈ D extending s, t, respectively, s ′ ρ t ′ , while there is no such relation for the pair s ↾ (ℓ − 1), t ↾ (ℓ − 1). Since an interesting level for a pair of nodes s, t predetermines the relations between any pair s ′ , t ′ extending s, t, respectively, passing numbers are unnecessary to the characterization.…”