$2^{\aleph_0}$ pairwise non-isomorphic maximal-closed subgroups of Sym$(\mathbb{N})$ via the classification of the reducts of the Henson digraphs

TL;DR

This paper classifies the reducts of Henson digraphs, showing there are continuum many non-isomorphic structures with no proper non-trivial reducts, and constructs many maximal-closed subgroups of Sym$( ats)$ answering a longstanding question.

Contribution

It provides a complete classification of reducts of Henson digraphs and constructs continuum many non-isomorphic maximal-closed subgroups of Sym$( ats)$.

Findings

There are $2^{eth_0}$ non-isomorphic Henson digraphs with no proper non-trivial reducts.

There exist $2^{eth_0}$ pairwise non-conjugate maximal-closed subgroups of Sym$( ats)$.

These groups are also non-isomorphic as abstract groups.

Abstract

Given two structures $\mathcal{M}$ and $\mathcal{N}$ on the same domain, we say that $\mathcal{N}$ is a reduct of $\mathcal{M}$ if all $\emptyset$-definable relations of $\mathcal{N}$ are $\emptyset$-definable in $\mathcal{M}$. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are $\aleph_0$-categorical, determining their reducts is equivalent to determining all closed supergroups $G<$ Sym$(\mathbb{N})$ of their automorphism groups. A consequence of the classification is that there are $2^{\aleph_0}$ pairwise non-isomorphic Henson digraphs which have no proper non-trivial reducts. Taking their automorphisms groups gives a positive answer to a question of Macpherson that asked if there are $2^{\aleph_0}$ pairwise non-conjugate maximal-closed subgroups of Sym$(\mathbb{N})$. By the reconstruction results of Rubin, these groups are also non-isomorphic as abstract groups.