Characterisation of the big Ramsey degrees of the generic partial order

TL;DR

This work determines the big Ramsey degrees of finite substructures of the generic partial order $\mathbf{P}$ by introducing poset-diaries, a tree-like combinatorial framework that encodes level-by-level events in a coding tree of 1-types. The authors prove that the big Ramsey degree of any finite poset $\mathbf{Q}$ in $\mathbf{P}$ equals $|T(\mathbf{Q})|\cdot |\mathrm{Aut}(\mathbf{Q})|$, and show that expansions arising from poset-diaries yield a big Ramsey structure for $\mathbf{P}$, connecting to topological dynamics via the universal completion flow. Upper bounds are obtained through refined Carlson–Simpson arguments on shape-preserving embeddings, while lower bounds are established via recurrent colourings that realize all labellings of $\mathbf{Q}$. This framework generalizes to other infinitary finite-relational contexts and clarifies how infinitary Ramsey phenomena extend Nešetřil–Rödl-type results beyond binary free-amalgamation classes. The results also illuminate connections to similar characterisations for linear orders and triangle-free graphs, suggesting a unified diary-like approach to big Ramsey theory in finite-arity relational languages. The techniques hold promise for broader applicability and potential extensions to higher arity and more complex amalgamation schemes.

Abstract

As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite partial orders endowed with linear extensions form a Ramsey class (this result was announced by Nešetřil and Rödl in 1984 with first published proof by Paoli, Trotter and Walker in 1985). Towards this, we refine earlier upper bounds obtained by Hubička based on a new connection of big Ramsey degrees to the Carlson-Simpson theorem and we also introduce a new technique of giving lower bounds using an iterated application of the upper-bound theorem.

Figures (7)

• Figure 1: Possible levels in poset-diaries.
• Figure 2: Diaries of $\mathbf {A}_2$.
• Figure 3: Poset-diaries of $\mathbf {C}_2$.
• Figure 4: Initial part of the tree of types of an enumerated linear order $(\mathbf {Q},\leq_\mathbf {Q})$ (left) and of the enumerated partial order $\mathbf {P}$ (right). The bold node on each level corresponds to the coding node.
• Figure 5: Function $f\circ g$.

Theorems & Definitions (63)

• Theorem 1.1: Hubička Hubicka2020CS
• Definition 1.2: Partial order $(\Sigma^*,\preceq)$
• Proposition 1.3: Hubicka2020CS
• proof : Proof of Proposition \ref{['prop:pos']}
• Definition 1.4: Partial orders $(\Sigma^*_\ell,\trianglelefteq)$
• Definition 1.5: Poset-diaries
• Example 1.6
• Theorem 1.7
• Example 1.8
• Theorem 2.1