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The famous Lovász Local Lemma was derived in the paper of P. Erdős and Lovász to prove that any _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least
_Δ(H) ≥ ¼ r_<sup>_n−1_</sup>.
A long series of papers is devoted to the improvement of this classical result for different classesof uniform hypergraphs.
In our work we deal with colorings of simple hypergraphs, i.e. hypergraphs in which everytwo distinct edges do not share more than one vertex. By using a multipass random recoloringwe show that any simple _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least
_Δ(H) ≥ с · nr_<sup>_n−1_</sup>
where _c_ > 0 is an absolute constant. We also give some applications of our probabilistic technique, we establish a new lower bound for the Van der Waerden number and extend the main result to the _b_-simple case.
The work of the second author was supported by Russian Foundation of Fundamental Research (grant № 12-01-00683-a), by the program “Leading Scientific Schools” (grant no. NSh-2964.2014.1) and by the grant of the President of Russian Federation MK-692.2014.1