  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
