Reducts of the random partial order

TL;DR

This paper classifies all structures first-order definable in the random partial order, revealing exactly five equivalence classes and connecting to broader conjectures about homogeneous structures.

Contribution

It provides a complete classification of first-order definable structures in the random partial order, identifying five symmetry classes and supporting a conjecture on finiteness.

Findings

Five equivalence classes of definable structures identified

Exactly five closed permutation groups contain the automorphism group

Supports conjecture on finiteness of definable structures in homogeneous structures

Abstract

We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that these structures fall into precisely five equivalence classes. We achieve this result by showing that there exist exactly five closed permutation groups which contain the automorphism group of the random partial order, and thus expose all symmetries of this structure. Our classification lines up with previous similar classifications, such as the structures definable in the random graph or the order of the rationals; it also provides further evidence for a conjecture due to Simon Thomas which states that the number of structures definable in a homogeneous structure in a finite relational language is, up to first-order interdefinability, always finite. The method we employ is based on a Ramsey-theoretic analysis of functions acting on the random partial order, which allows us to find patterns in such functions and make them accessible to finite combinatorial arguments.