Van der Waerden's theorem; the Erdős–Turán conjecture / Szemerédi's theorem (without proof); characters of a finite Abelian group and the Pontryagin dual; Fourier transform over finite Abelian groups; Parseval's identity / Plancherel's formula

Fourier transforms of subsets of finite Abelian groups; Roth's theorem in ℤ₃^m (Meshulam's theorem) with proof; proof of Roth's theorem in {1, ..., n}

Progression of upper bounds in Roth's theorem; Freiman isomorphisms; Behrend's construction of large subsets of {1, ..., n} with no 3-term APs; the triangle removal lemma; derivation of Roth's theorem from the triangle removal lemma; the "corners" theorem in [n]² and derivation of Roth's theorem from it; the K_r+1^(r) removal lemma; the "corners" theorem in [n]^d

The notion of ε-regularity; Szemerédi's regularity lemma; the triangle counting lemma; proof of the triangle removal lemma; proof of Szemerédi's regularity lemma

Proof of Szmerédi's regularity lemma (continued); the upper bound on the size of 3AP-free sets in (ℤ_q)^m due to Ellenberg–Gijswijt, after Croot–Lev–Pach

Sum-free sets; largest sum-free subsets of {1, ..., n}; every n-element set of integers contains a sum-free subset with (n+1)/3 elements; the theorem of Eberhard, Green, and Manners (without proof); infinite sum-free sets either contain only odd numbers or have upper density at most 2/5 (Łuczak); Sidon sets in {1, ..., n}: the upper bound of Erdős and Turán and the construction of Singer / Ruzsa

B_h-sets in {1, ..., n}: the trivial upper bound and the construction of Bose and Chowla; the Cauchy–Davenport theorem; Kneser's theorem (without proof); the sum-product conjecture of Erdős and Szemerédi; the upper bound of Erdős and Szemerédi; progression of lower bounds; the argument of Elekes; the Szemerédi–Trotter incidence theorem

the crossing lemma of Ajtai–Chvátal–Newborn–Szemerédi; Solymosi's bound; the Littlewood–Offord problem; Sperner's theorem and Erdős' bound

singularity of random Bernoulli matrices; Halász' theorem

Frieman's 3k-4 theorem (without proof); the Plünnecke–Ruzsa inequality (estimates for the growth of iterated sumsets); geometry of numbers: lattices, fundamental parallelepipeds, determinants; Blichfeldt's lemma

Minkowski's first theorem; Minkowski's second theorem (without proof); Bohr neighbourhoods contain large generalised APs; Bogolyubov's theorem

Ruzsa's representation lemma; Freiman's inverse theorem (with proof)