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Are there any introductory texts available on mathematical recursion theory (i.e., not aimed at computer scientists) that include a good amount of applications to mathematical combinatorics or other areas of discrete math? For example, are there any that draw explicit connections with recurrence relations in combinatorics?

I haven't had much luck searching online.

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    I doubt it - recursion theory is usually only concerned with what is not computable, and most problems in combinatorics are at worst "difficult" to compute. You might see more success with the more modern term "computability theory" rather than "recursion theory", but I don't think there's much out there. – 2017-02-21
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    Ditto what Reese said, but whatever it is you're precisely looking for, Concrete Mathematics by Knuth, at el, https://en.wikipedia.org/wiki/Concrete_Mathematics might come close. – 2017-02-21
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    @Reese How about recursion/computability theory books including applications to math in general? A famous example is Diophantine equations in Hilbert's Tenth Problem. for example, so maybe number theory examples? It's easy to find model theory books aimed at algebra and set theory books aimed at real analysis. Are there any analogues for recursion theory you could think of? – 2017-02-21
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    @CuriousKid7 To be honest, Hilbert's Tenth is the only real example of that I know of. There's a lot of work done in computable structure theory which relates to number theory and other fields, but I'm not sure it's quite what you're looking for (computable structure theory asks questions like "are there two computable groups that are both isomorphic to $(\mathbb{Z}, +)$ but have no computable isomorphism between them"). The best text I know for that is Ash and Knight's "Computable Structures and the Hyperarithmetic Hierarchy", but it's far from introductory. – 2017-02-21
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    @Reese: many classical combinatorial theorems have computable instances with no computable solutions - Ramsey's theorem, Hindman's theorem, and versions of the stable marriage theorem are examples of this phenomenon. – 2017-02-21

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There has been a great deal of work on combinatorics in computability theory. However, there is little on the subject in "introductory" books, such as books aimed at undergraduates or at readers who don't know the basics of computability theory already.

One recent graduate-level book on the subject is Slicing the Truth by Denis Hirschfeldt. He has an expository article with most of the content online as well.

Most of the work in this area is not in books, however, and is simply in research papers. A famous and exceptional older paper is "Ramsey's Theorem and Recursion Theory" by Carl Jockusch (J. Symb. Logic, 1972). That might be a good starting point, for a reader who knows the basics of computability theory (such as the beginning parts of Soare's book). More modern papers worth reading include:

  • "Logical analysis of some theorems of combinatorics and topological dynamics" by Blass, Hirst, and Simpson (Contemp. Math. 65, 1987)
  • "On the strength of Ramsey's theorem for pairs" by Cholak, Jockusch, and Slaman (J. Symb. Logic, 2001)
  • "On uniform relationships between combinatorial problems" by Dorais, Dzhafarov, Hirst, Mileti, and Shafer (Trans. AMS, 2016).
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    Thank you! Slicing the Truth looks really interesting and exactly what I was looking for. I'll try to work my way up to it. (Also, the book Logical Number Theory by Smorynski is pretty good). – 2017-02-22