Reverse mathematics and infinite traceable graphs

TL;DR

This paper analyzes the proof strength of certain combinatorial principles in graph and lattice theory using reverse mathematics, establishing equivalences with Ramsey's theorem for 4-tuples and proving some statements within base systems.

Contribution

It demonstrates the logical strength of specific graph theory and lattice theory statements in the framework of reverse mathematics, clarifying their relation to Ramsey's theorem.

Findings

Graph theory statements are equivalent to Ramsey's theorem for 4-tuples over RCA.

Lattice theory statement is provable in RCA.

Provides insight into the proof-theoretic strength of combinatorial principles.

Abstract

This paper falls within the general program of investigating the proof theoretic strength (in terms of reverse mathematics) of combinatorial principals which follow from versions of Ramsey's theorem. We examine two statements in graph theory and one statement in lattice theory proved by Galvin, Rival and Sands \cite{GRS:82} using Ramsey's theorem for 4-tuples. Our main results are that the statements concerning graph theory are equivalent to Ramsey's theorem for 4-tuples over $\RCA$ while the statement concerning lattices is provable in $\RCA$. Revised 12/2010. To appear in Archive for Mathematical Logic