# 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…

## 187 Citations

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We say that a graph $G$ is Ramsey for $H_1$ versus $H_2$, and write $G \to (H_1,H_2)$, if every red-blue colouring of the edges of $G$ contains either a red copy of $H_1$ or a blue copy of $H_2$. In…

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A celebrated result of Rödl and Ruciński states that for every graph $F$ , which is not a forest of stars and paths of length 3, and fixed number of colours $r\geqslant 2$ there exist positive…

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