We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several examples of colorings of the integers which do not have computable witnesses to Hindman's Theorem.
"A Simple Proof and Some Difficult Examples for Hindman's Theorem." Notre Dame J. Formal Logic 53 (1) 53 - 65, 2012. https://doi.org/10.1215/00294527-1626518