## Combinatorial theorems in sparse random sets.(English)Zbl 1351.05204

Summary: We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán’s theorem, Szemerédi’s theorem and Ramsey’s theorem, hold almost surely inside sparse random sets. For instance, we extend Turán’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 probability at least $$C n^{-2/(t+1)}$$, then, with probability tending to 1 as $$n$$ tends to infinity, every subgraph of $$G$$ with at least $$\left(1 - \frac{1}{t-1} + \epsilon\right) e(G)$$ edges contains a copy of $$K_t$$. This is sharp up to the constant $$C$$. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Turán theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by M. Schacht [Ann. Math. (2) 184, No. 2, 333–365 (2016; Zbl 1351.05207)] and by E. Friedgut et al. [Random Struct. Algorithms 37, No. 4, 407–436 (2010; Zbl 1228.05284)].

### MSC:

 05C80 Random graphs (graph-theoretic aspects) 05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.) 05C42 Density (toughness, etc.) 05D10 Ramsey theory

### Citations:

Zbl 1228.05284; Zbl 1351.05207
Full Text:

### References:

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