For people interested in arithmetic progression type theorems, one of my favorite theorems along those lines (though technically not actually about arithmetic p

For people interested in arithmetic progression type theorems, one of my favorite theorems along those lines (though technically not actually about arithmetic progressions, but extremely closesly related), and a very under-appreciated gem, is Hindman's Theorem. Hindman's Theorem says that if you color the natural numbers with finitely many colors, there must be some infinite subset D of the natural numbers such that every finite sum of elements of D has the same color. The proof, asontonishingly, is an application of free ultrafilters, and is actually simple enough that someone with an advanced undergrad background in pure math can understand it with just a few hours of reading (e.g. see [1]) [1] https://web.williams.edu/Mathematics/lg5/Hindman.pdf https://web.williams.edu/Mathematics/lg5/Hindman.pdf