Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.
Henry Towsner. "Hindman's theorem: an ultrafilter argument in second order arithmetic." J. Symbolic Logic 76 (1) 353 - 360, March 2011. https://doi.org/10.2178/jsl/1294171005