Combinatorial theorems relative to a random set

We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

Graph removal lemmas

The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.Comment: 35 page

Extremal results in sparse pseudorandom graphs

Szemer\'edi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemer\'edi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.Comment: 70 pages, accepted for publication in Adv. Mat

What is good mathematics?

Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer\'edi's theorem is presented.Comment: 12 pages, no figures. To appear, Bull. Amer. Math. So