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.
"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