Binary simple homogeneous structures are supersimple with finite rank

TL;DR

This paper proves that infinite simple homogeneous structures with finite relational vocabulary and arity at most 2 are supersimple with finite SU-rank, bounded by the number of 2-types over the empty set.

Contribution

It establishes a new link between simplicity, homogeneity, and supersimplicity with finite SU-rank in binary structures.

Findings

Infinite simple homogeneous structures with finite vocabulary are supersimple.

SU-rank of such structures is finite and bounded by the number of 2-types.

Provides a characterization of the complexity of these structures.

Abstract

Suppose that M is an infinite structure with finite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous then its complete theory is supersimple with finite SU-rank which cannot exceed the number of complete 2-types over the empty set.