Geodesic
Extremal problems for colourings of uniform hypergraphs
Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1253-1290

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We study a classical problem (first posed by Erdős) in the extremal theory of hypergraphs. According to Erdős, a hypergraph possesses property $B$ if its set of vertices admits a 2-colouring such that no edge of the hypergraph is monochromatic. The problem is to find the minimum $m(n)$ of all $m$ such that there is an $n$-uniform (each edge contains exactly $n$ vertices) hypergraph with exactly $m$ edges that does not possess property $B$. We consider more general problems (including the case of polychromatic colourings) and introduce a number of parametric properties of hypergraphs. We obtain estimates for analogues of $m(n)$ for extremal problems on various classes of hypergraphs. 
@article{IM2_2007_71_6_a6,
     author = {D. A. Shabanov},
     title = {Extremal problems for colourings of uniform hypergraphs},
     journal = {Izvestiya. Mathematics },
     pages = {1253--1290},
     publisher = {mathdoc},
     volume = {71},
     number = {6},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2007_71_6_a6/}
}
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D. A. Shabanov. Extremal problems for colourings of uniform hypergraphs. Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1253-1290. http://geodesic.mathdoc.fr/item/IM2_2007_71_6_a6/