There are many proofs of various lower bounds on the usual Ramsey numbers. Combinatorialists have been concerned mainly with two issues, obtaining better bounds, and considering constructive versus non-constructive bounds. I will consider some logical aspects, beginning to answer the question: What is the difference between these bounds in terms of the axioms required to prove them? This sort of work fits in with the program of Reverse Mathematics, though here we work with the weaker theories of bounded arithmetic, since they are a more suitable context for finite combinatorics.