Symmetries of structures that fail to interpret something finite

TL;DR

This paper explores the algebraic and structural properties of directed graphs that do not interpret all finite structures, revealing new invariance properties and their implications for constraint satisfaction problems.

Contribution

It generalizes existing theorems on interpretability in graphs and introduces new algebraic invariance properties relevant to both finite and infinite structures.

Findings

Generalized several theorems on interpretability in graphs

Identified new algebraic invariance properties for such graphs

Constructed a countably categorical hypergraph lacking certain interpretability features

Abstract

We investigate structural implications arising from the condition that a given directed graph does not interpret, in the sense of primitive positive interpretation with parameters or orbits, every finite structure. Our results generalize several theorems from the literature and yield further algebraic invariance properties that must be satisfied in every such graph. Algebraic properties of this kind are tightly connected to the tractability of constraint satisfaction problems, and we obtain new such properties even for infinite countably categorical graphs. We balance these positive results by showing the existence of a countably categorical hypergraph that fails to interpret some finite structure, while still lacking some of the most essential algebraic invariance properties known to hold for finite structures.