Sharp thresholds for graph properties

OpenAI's proofs (2026) of the Friedgut–Kalai conjectures: increasing graph properties of G(n,p) have threshold width O(log(1/ε)/(log n)²), and r-uniform hypergraph properties width O((log n)^(−r/(r−1))). The course develops biased Fourier analysis and hypercontractivity, restricts symmetric functions to the edges meeting a random block of vertices, and captures each Fourier support at one vertex.

Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Biased Fourier analysis and hypercontractivity
2. Vertex blocks and graph properties
3. Hypergraph properties

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Finite probability, Hölder's inequality and inner products on finite-dimensional spaces.