The Henson graphs: colorings and codings

TL;DR

This paper explores the finite big Ramsey degrees of finite graphs within Henson graphs and constructs a computable coloring that encodes the halting set, linking combinatorics with computability theory.

Contribution

It demonstrates the existence of a computable coloring in Henson graphs that encodes the halting set, extending the understanding of Ramsey degrees and computability.

Findings

Finite big Ramsey degrees are exactly characterized for Henson graphs.

A computable coloring encodes the halting set within Henson graphs.

The result links combinatorial properties of graphs with computability theory.

Abstract

By recent work of \citet{DobrinenICM} and \citet{Balko7} we know that every finite $G$ in the Henson graph $\mathbb{H}_{n+1}$ (the universal ultrahomogeneous $(n+1)$-clique free graph) has exact finite big Ramsey degree $k({G,n})$. That is, there is a positive integer $k({G,n})$ such that for each finite coloring $C$ of the copies of $G$ in $\mathbb{H}_{n+1}$, there is $\tilde{\mathbb{H}}$, a substructure of $\mathbb{H}_{n+1}$ and isomorphic to $\mathbb{H}_{n+1}$, such that in $\tilde{\mathbb{H}}$ at most $k({G,n})$ colors are used on the copies of $G$ in $\tilde{\mathbb{H}}$. Moreover, for exactness, for some coloring and all corresponding $\tilde{\mathbb{H}}$, all $k({G,n})$ colors are needed. The ultimate result here is that if $|G|\geq 2$, then there is a finite computable coloring $C$ such that, for all such $\tilde{\mathbb{H}}$, we have that $\tilde{\mathbb{H}}$ computes $\emptyset^{(|G|-1)}$ (and hence the halting set).