Geodesic
Van der Waerden's function and colourings of hypergraphs
Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 1063-1091

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A classical problem of combinatorial number theory is to compute van der Waerden's function $W(n,r)$. Using random colourings of hypergraphs, we get a new asymptotic lower bound for $W(n,r)$ which improves previous results for a wide range of values of $n$ and $r$.
Keywords: van der Waerden's theorem, arithmetic progressions, hypergraph, chromatic number. 
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     author = {D. A. Shabanov},
     title = {Van der {Waerden's} function and colourings of hypergraphs},
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D. A. Shabanov. Van der Waerden's function and colourings of hypergraphs. Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 1063-1091. http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a8/