## Dynamic concentration of the triangle-free process

#### journal contribution

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The triangle-free process begins with an empty graph on *n* vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on Ramsey numbers: we show *R*(3,*t*) > (1/4 − *o*(1))*t* 2/ log *t*, which is within a 4 + *o*(1) factor of the best known upper bound. Furthermore, we determine which bounded size subgraphs are likely to appear in the maximal triangle-free graph produced by the triangle-free process: they are precisely those triangle-free graphs with maximal average density at most 2.