Math 873: Advanced Topics in Foundations - Reverse Mathematics of Combinatorial Principles
Math 770 and 773
Reverse mathematics tries to pinpoint the exact axioms needed to prove a theorem in "ordinary" mathematics, ideally by showing that the proof can be "reversed" and a certain set of axioms can be proved from the theorem (over a weaker base theory, usually RCA_0, the theory corresponding to "computable mathematics").
Most theorems in analysis and algebra reverse to one of just five "big" axiom systems. Combinatorics, however, yields a large number of new axiom systems, and the relationship between these has been the subject of intense investigation for several decades.
I plan to start with the seminal 1972 paper by Jockusch on computable combinatorics and continue with highlights like Seetapun's 1995 separation of Ramsey's Theorem for Pairs from the axioms system ACA_0, the Cholak/Jockusch/Slaman 2001 paper expanding Seetapun's result in a number of ways, the 2007 and 2009 papers by Hirschfeldt/Shore and Hirschfeldt/Slaman/Shore on order-theoretic principles and theorems from model theory, and the recent papers by Liu (showing that Ramsey's Theorem for Pairs does not imply Weak König's Lemma), by Chong/Slaman/Yang (that the stable version of Ramsey's Theorem does not imply the full version) and by Wang (that Thin Set Theorem implies almost no other combinatorial principle).