# Coding in graphs and linear orderings

## TL;DR

This paper explores the limits of interpreting graphs within linear orderings using computable formulas, demonstrating that certain embeddings cannot be uniformly characterized by fixed formulas, highlighting fundamental differences in interpretability.

## Contribution

It proves that no fixed tuple of $L_{,}$ formulas can uniformly interpret all graphs in linear orderings, extending understanding of interpretability limitations.

## Key findings

- Directed graphs can be embedded into undirected graphs with a uniform interpretation.
- Some graphs are not Medvedev reducible to any linear ordering.
- No fixed formulas can interpret all graphs within linear orderings.

## Abstract

There is a Turing computable embedding $\Phi$ of directed graphs $A$ in undirected graphs. Moreover, there is a fixed tuple of formulas that give a uniform interpretation; i.e., for all directed graphs $A$, these formulas interpret $A$ in $\Phi(G)$. It follows that A is Medvedev reducible to $\Phi(A)$ uniformly; i.e., there is a fixed Turing operator that serves for all $A$. We observe that there is a graph $G$ that is not Medvedev reducible to any linear ordering. Hence, $G$ is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable $\Sigma_2$ formulas. Any graph can be interpreted in a linear ordering using computable $\Sigma_3$ formulas. Friedman and Stanley gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of $L_{\omega_1,\omega}$ formulas that, for all $G$, interpret the input graph $G$ in the output linear ordering $L(G)$. Harrison-Trainor and Montalb\'an have also shown this, by a quite different proof.