Random and exact structures in combinatorics

Abstract

In this thesis I aim to show several developments related to notions of randomness and structure in combinatorics and probability. One central notion, the pseudorandomness-structure dichotomy, has played a key role in additive combinatorics and extremal graph theory. More generally, however, such notions come into play in the study of combinatorial probability and the use of random processes in extremal combinatorics. In a broader view, randomness (and the pseudorandomness notions which resemble it along various axes) can be viewed as a type of structure in and of itself which has certain typical and global properties that may be exploited to exhibit or constrain combinatorial and probabilistic behavior.

These broader ideas often come in concert to allow the construction or extraction of exact behavior. I have chosen three directions along which to study this theme: the singularity of discrete random matrices, thresholds for Steiner triple systems, and improved bounds for Szemerédi's theorem. Each concerns breakthroughs in central questions of the fundamental areas of random matrices, combinatorial designs, and additive combinatorics.