Ordered Variable Word theorem does not imply RT^2_2, admits low2 computable solutions, and yields Carlson-Simpson and Open Dual Ramsey over ACA0.
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Countable structures admitting finite monomorphic decomposition have finite big Ramsey degrees iff their monomorphic parts do, via chainability and Laver's theorem on Fraïssé's conjecture.
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The reverse mathematics of the Ordered Variable Word theorem
Ordered Variable Word theorem does not imply RT^2_2, admits low2 computable solutions, and yields Carlson-Simpson and Open Dual Ramsey over ACA0.
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Fra\"{i}ss\'{e}'s Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition
Countable structures admitting finite monomorphic decomposition have finite big Ramsey degrees iff their monomorphic parts do, via chainability and Laver's theorem on Fraïssé's conjecture.