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Colors of the Pseudotree

David Chodounský, Monroe Eskew, Thilo Weinert

Abstract

We investigate big Ramsey degrees of finite substructures of the universal countable homogeneous meet-tree and its binary variant. We prove that structures containing antichains have infinite big Ramsey degrees, and the big Ramsey degree of a 2-element chain is at least 8 and 7 for the binary variant. We deduce that the generic C-relation does not have finite big Ramsey degrees.

Colors of the Pseudotree

Abstract

We investigate big Ramsey degrees of finite substructures of the universal countable homogeneous meet-tree and its binary variant. We prove that structures containing antichains have infinite big Ramsey degrees, and the big Ramsey degree of a 2-element chain is at least 8 and 7 for the binary variant. We deduce that the generic C-relation does not have finite big Ramsey degrees.

Paper Structure

This paper contains 6 sections, 9 theorems, 1 equation.

Table of Contents

  1. Big Ramsey degrees
  2. The structure of universal homogeneous pseudotrees
  3. Indivisibility
  4. A coloring of chains
  5. The oscillation on antichains
  6. The generic $C$-relation

Key Result

Proposition 2

The pseudotrees $\psi_\omega$ and $\psi_2$ can be up to an isomorphism characterized as a countable pseudotree with the following two properties.

Theorems & Definitions (17)

  • Definition 1
  • Proposition 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Proposition 6
  • Claim
  • Theorem 7
  • proof
  • Claim
  • ...and 7 more