21 Oct 2016: Counting uses of a theorem

Jeff Hirst

Many proofs use a bootstrapping process. Initially, a simple version is proved, and then extensions are proved by repeated applications of the simple result. Proofs of many results about graph colorings use this strategy. We will concentrate on a specific example. Suppose we write RT(2,n) for Ramsey’s theorem for pairs and n colors, a graph coloring theorem. (Pictures will be provided.) The usual proof of RT(2,4) uses two applications of RT(2,2). In this talk, we address the question: Can RT(2,4) be proved with one use of RT(2,2)? Of course, the answer depends on the base system chosen and the formalization of what constitutes a use. In some settings, a formalization of Weihrauch reductions can lend insight, and the law of the excluded middle plays a surprising role.