Here is a link to Uri Leron’s paper Structuring Mathematical Proofs, The American Mathematical Monthly 90 (3), (Mar., 1983), 174-185. Dr. Leron talks here about the non-linear nature of proofs (remember the examples I mentioned from Szemeredi and Shelah) and discusses what he calls the “structural method.” It is worth keeping in mind his ideas as you continue through graduate school, and especially when faced with the tasks of giving a talk or writing up your results (even if you disagree with him).

The basic idea of the structural method is that proofs should perhaps be presented in levels, each giving at least an outline of a complete argument. As you descend through the levels, you fill in details. It is a fairly natural approach (like dividing a result into a series of lemmas), and it has the advantage that it helps the audience understand where the argument is going and have a better global picture of what is going on. It is also harder than one would think, in actual practice, to organize one’s arguments according to this method and, even if just for practice, I find it useful every now and then to see how to present a result, even one whose proof I understand fairly well, following this approach rather than the more standard linear technique.