Amalgamation Classes with $\exists$-Resolutions

TL;DR

This paper investigates amalgamation classes of finite graphs with distance-preserving embeddings, focusing on classes with $orall$-resolutions, and explores the existence of certain injective structures without finite closures, addressing Moss's conjecture.

Contribution

It extends the study of amalgamation classes with $orall$-resolutions and analyzes conditions limiting the existence of injective structures without finite closures, relating to Moss's conjecture.

Findings

The question is more interesting in classes with $orall$-resolutions.

Conditions are provided that limit the existence of injective structures without finite closures.

The paper discusses the implications for Moss's conjecture in this context.

Abstract

Let $K_d$ denote the class of all finite graphs and, for graphs $A\subseteq B$, say $A \leq_d B$ if distances in $A$ are preserved in $B$; i.e. for $a, a' \in A$ the length of the shortest path in $A$ from $a$ to $a'$ is the same as the length of the shortest path in $B$ from $a$ to $a'$. In this situation $(K_d, \leq_d)$ forms an amalgamation class and one can perform a Hrushovski construction to obtain a generic of the class. One particular feature of the class $(K_d, \leq_d)$ is that a closed superset of a finite set need not include all minimal pairs obtained iteratively over that set but only enough such pairs to resolve distances; we will say that such classes have $\exists$-resolutions. Larry Moss conjectured the existence of graph $M$ which was $(K_d, \leq_d)$-injective (for $A \leq_d B$ any isometric embedding of $A$ into $M$ extends to an isometric embedding of $B$ into $M$) but without finite closures. We examine Moss's conjecture in the more general context of amalgamation classes. In particular, we will show that the question is in some sense more interesting in classes with $\exists$-resolutions and will give some conditions under which the possibility of such structures is limited.