TITLE: Applications of Measure, Integration and Banach Spaces to Combinatorics
QUESTION [17 upvotes]: 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.

REPLY [3 votes]: As warm-up exercises using Cauchy-Schwarz and Holder's inequality, you could mention restricted subgraph bounds and incidence bounds. When you rearrange these bounds to bound the number of "rich/popular" points or vertices you get (weak) $L^p$ bounds for the degree of each vertex (viewed a function on the vertex set). In this setting it's easy to see how interpolation works, and it gives you a feel for how $L^p$ spaces work in a probability setting (i.e. on a compact domain). 
Here's a reference for the restricted subgraph bounds: http://murphmath.wordpress.com/2012/06/19/restricted-subgraph-bounds/. Instead of the convexity of ${x\choose s}$ in $x$, you can use Holder's inequality. The bounds can be rearranged using Chebyshev's inequality to state that (in the notation of that blog post):
$$|\{b\in B\colon \mathrm{deg}(b)\geq t\}|\leq C \frac{|A|^s}{t^s}$$
where $C$ is a constant depending on $s$ and $t$. Then using the "layer-cake" theorem (which is a nice exercise using Fubini's theorem!) you get that
$$||\mathrm{deg}(b)||_{L^s(B)}^s \leq C'|A|^s\log|A|.$$
I know this is pretty trivial, but this is actually what's at stake in the Kakeya maximal conjecture. Here one has a collection of tubes $T_1,\ldots,T_n$ and the goal is to prove $L^p$ bounds for the function
$$ f(x)=\sum_{i=1}^n \chi_{T_i}(x),$$
where $\chi_{T_i}$ is the indicator function on the tube $T_i$. Thus we are seeking $L^p$ bounds for the function that counts how many tubes are incident to to a point $x$! The proof of the two dimensional Kakeya conjecture is essentially the same as the proof restricted subgraph bound for $s=2$ in that blog post---the incidence graph for points and lines in a plane contains no $K_{2,2}$'s, and modulo some technicalities in the Kakeya case, the main tool is Cauchy-Schwarz. In fact, Wolff's paper on the Kakeya problem contains a finite field analog where the proof is exactly by Cauchy-Schwarz.