  An $n$-vertex graph is called $C$-Ramsey if it has no clique or independent
set of size $C\log_2 n$ (i.e., if it has near-optimal Ramsey behavior). In this
paper, we study edge-statistics in Ramsey graphs, in particular obtaining very
precise control of the distribution of the number of edges in a random vertex
subset of a $C$-Ramsey graph. This brings together two ongoing lines of
research: the study of "random-like" properties of Ramsey graphs and the study
of small-ball probabilities for low-degree polynomials of independent random
variables.
  The proof proceeds via an "additive structure" dichotomy on the degree
sequence, and involves a wide range of different tools from Fourier analysis,
random matrix theory, the theory of Boolean functions, probabilistic
combinatorics, and low-rank approximation. One of the consequences of our
result is the resolution of an old conjecture of Erd\H{o}s and McKay, for which
Erd\H{o}s offered one of his notorious monetary prizes.
