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Structure and randomness in combinatorics
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Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable to decompose such objects into a \emph{structured} component, a \emph{pseudo-random} component, and a \emph{small} component (i.e. an error term); in many cases it is the structured component which then dominates. We illustrate this philosophy in a number of model cases.
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Quantitative analytic stable regularity
Stable real-valued functions, defined by omitting ladders, admit definable partitions and equipartitions with polynomial-in-1/ε many parts such that every pair is almost constant.
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