I'm in my last year of an undergraduate degree in pure maths, doing my thesis on probabilistic combinatorics. In Australia (at least at my university), Hungarian-style combinatorics is classified as "pure".
I'm currently thinking about PhDs and where to apply. One thing that I've noticed is that combinatorics is often classified under "applied maths" (for example, this is the case at MIT).
One important reason I'd consider pursuing a PhD at an American university is for the coursework component -- I'd like to broadly improve my foundational understanding of mathematics, and I like the possibility of changing my area of focus after deeper exposure to different areas of mathematics. However, I have comparatively little interest in applied mathematics coursework (other than combinatorics, of course).
I have a submitted paper in combinatorics, my undergraduate thesis will be in combinatorics, and by far my strongest reference will be from a combinatorialist. Other than my supervisor, my strongest potential references would come from areas that everyone seems to agree are "pure". The higher-level classes in my transcript will also be almost entirely "pure".
For American universities which have both a pure and an applied math program, which one does one typically choose to study combinatorics? Is it flexible? With a cursory search, I wasn't able to find a university quite so explicit about it as MIT.
For universities that classify combinatorics as "applied":
Given my situation, would my application be stronger to pure or applied maths?
Can one do a PhD in pure mathematics with the (tentative) intention for the research component to actually be "applied" (combinatorics)?
Can one do a PhD in applied mathematics but actually take mostly pure coursework?