Structures preserved by primitive actions of $S_\omega$

TL;DR

This paper establishes a dichotomy for structures preserved by primitive actions of the infinite symmetric group, classifying their computational complexity for the constraint satisfaction problem.

Contribution

It introduces a classification theorem linking primitive actions of $S_\omega$ to the complexity of associated CSPs, based on properties of first-order reducts of Johnson graphs.

Findings

Structures either primitively positively interpret all finite structures or have polynomial-time CSPs.

The classification hinges on properties of automorphism groups and Ramsey expansions.

The results connect group actions with computational complexity of logical structures.

Abstract

We present a dichotomy for structures $A$ that are preserved by primitive actions of $S_{\omega} = \text{Sym}({\mathbb N})$: such a structure primitively positively constructs all finite structures and the constraint satisfaction problem is NP-complete, or the constraint satisfaction problem for $A$ is in P. To prove our result, we study the first-order reducts of the Johnson graph $J(k)$, for $k \geq 2$, whose automorphism group $G$ equals the action of $\text{Sym}({\mathbb N})$ on the set $V$ of $k$-element subsets of $\mathbb N$. We use the fact that $J(k)$ has a finitely bounded homogeneous Ramsey expansion and that $G$ is a maximal closed subgroup of $\text{Sym}(V)$.