The Henson graphs: colorings and codings
Peter Cholak, Natasha Dobrinen, Charlie McCoy

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 in the Henson graph (the universal ultrahomogeneous -clique free graph) has exact finite big Ramsey degree . That is, there is a positive integer such that for each finite coloring of the copies of in , there is , a substructure of and isomorphic to , such that in at most colors are used on the copies of in . Moreover, for exactness, for some coloring and all corresponding , all colors are needed. The ultimate result here is that if , then there is a finite computable coloring such that, for all such , we have that computes…
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