Reducts of the random partial order

Our result lines up with previous similar classifications for the random graph and 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. In the proof we use the new technique of “canonical functions” originally invented in the context of theoretical computer science, which allows for a systematic Ramsey-theoretic analysis of functions acting on the random partial order. The technique identifies patterns in arbitrary functions on the random partial order, which makes them accessible to finite combinatorial arguments.