Counting, graphs and extremal structures, where small certificates often settle big questions.
How large can a discrete structure be without containing a forbidden pattern? Ramsey theory, Turán problems and set systems, attacked mostly by random constructions, entropy and iterative algorithms.
Minors, tree-width and the Robertson–Seymour theory; the extremal density of K_t-minor-free graphs; and colouring by excluded minors, from the Four Colour Theorem to Hadwiger's conjecture.