Structure and randomness in combinatorics

TL;DR

This paper discusses a decomposition approach in combinatorics that separates large objects into structured, pseudo-random, and small components, highlighting the dominance of the structured part in many cases.

Contribution

It illustrates the decomposition methodology in combinatorics, emphasizing the importance of structured components over pseudo-random parts in various models.

Findings

Decomposition into structured, pseudo-random, and small parts is effective.

Structured components often dominate in combinatorial models.

The approach aids in understanding complex combinatorial objects.

Abstract

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.