A monotone graph property is a family of graphs which is closed under taking subgraphs (for example, the set of graphs that are K_4-free and C_7-free.) Erd\H{o}s, Frankl and R\"odl proved various results, and their work was continued by Pr\"omel and Steger. These results imply that for every monotone property there is a natural number k such that the size of the property is 2^{(1-1/k+o(1))n^2}. Further, in a series of papers, Pr\"omel and Steger showed that the error term can be sharpened for some natural properties. Recently, with Bollob\'as and Simonovits, we managed to extend these results to all monotone properties. In particular we have given essentially sharp estimates: for all monotone properties, there is an integer k and a positive constant c such that the cardinality of a property is between 2^{(1-1/k){n\choose 2}} and 2^{(1-1/k){n\choose 2}+n^{2-c}}. Our results are proved by applying techniques of probabilistic combinatorics, making use of a number of classical combinatorial theorems including Szemer\'edi's Regularity Lemma.

This is joint work with B\'ela Bollob\'as and Mikl\'os Simonovits.