Combinatorial theorems in sparse random sets

@article{Conlon2010CombinatorialTI,
  title={Combinatorial theorems in sparse random sets},
  author={David Conlon and William T. Gowers},
  journal={arXiv: Combinatorics},
  year={2010}
}
We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Tur\'an's theorem to the random setting by showing that for every $\epsilon > 0$ and every positive integer $t \geq 3$ there exists a constant $C$ such that, if $G$ is a random graph on $n$ vertices where each edge is chosen independently with… 
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