I'm going to be teaching a Master's level analysis course(measure theory, Lebesgue integration, Banach and Hilbert spaces, and if there's time, some spectral or PDE stuff) in the fall. My problem is that while roughly half of the students will actually be using analysis in their further work, the rest of them are going to specialize in combinatorics, and while I want to convince them that they should know this stuff as part of their general mathematical culture, I'd also like to try to connect it to what they'll be working on. So far, I've found survey articles on applications of Ramsey theory to Banach spaces, and applications of harmonic analysis to additive number theory, but I was wondering whether anyone had some suggestions of references for applications of classical analysis to old-fashioned, classical combinatorics. (I realise that this is a pretty tall order, as on many levels, these two fields are at antipodes.)
PS: I'm planning on talking about probability measures on discrete spaces, but I don't think that will convince the combinatorics people that hacking through the construction of the Lebesgue integral could have a practical payoff someday for them.