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. 
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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/

[1] P. Erdös, “On a combinatorial problem. I”, Nordisk Mat. Tidskr., 11 (1963), 5–10 | MR | Zbl

[2] P. Erdös, “On a combinatorial problem. II”, Acta Math. Acad. Sci. Hungar., 15:3–4 (1964), 445–447 | DOI | MR | Zbl

[3] J. Beck, “On 3-chromatic hypergraphs”, Discrete Math., 24:2 (1978), 127–137 | DOI | MR | Zbl

[4] J. Radhakrishnan, A. Srinivasan, “Improved bounds and algorithms for hypergraph two-coloring”, Random Structures Algorithms, 16:1 (2000), 4–32 | 3.0.CO;2-2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[5] N. Alon, J. H. Spencer, The probabilistic method, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 2000 | MR | Zbl

[6] D. A. Shabanov, “Ob odnoi kombinatornoi zadache Erdesha”, Dokl. RAN, 396:2 (2004), 166–169 | MR | Zbl

[7] P. Erdös, Ch. Ko, R. Rado, “Intersection theorems for systems of finite sets”, Quart. J. Math. Oxford, Ser. (2), 12 (1961), 313–320 | DOI | MR | Zbl

[8] R. Ahlswede, L. H. Khachatrian, “The complete nontrivial-intersection theorem for systems of finite sets”, J. Combin. Theory, Ser. A, 76:1 (1996), 121–138 | DOI | MR | Zbl

[9] R. Ahlswede, L. H. Khachatrian, “The complete intersection theorem for systems of finite sets”, European J. Combin., 18:2 (1997), 125–136 | DOI | MR | Zbl

[10] M. Deza, P. Frankl, “Erdös–Ko–Rado theorem – 22 years later”, SIAM J. Algebraic Discrete Methods, 4:4 (1983), 419–431 | DOI | MR | Zbl

[11] D. A. Shabanov, “O raskraskakh gipergrafov”, Dokl. RAN, 402:5 (2005), 605–608 | MR

[12] V. Feller, Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 1, Mir, M., 1984 ; W. Feller, An introduction to probability theory and its applications, Vol. I, Wiley, New York–London–Sydney, 1968 | MR | Zbl | MR | Zbl

[13] A. M. Raigorodskii, “Sistemy obschikh predstavitelei”, Fundam. i prikl. matem., 5:3 (1999), 851–860 | MR | Zbl

[14] N. N. Kuzyurin, “Asimptoticheskoe issledovanie zadachi o pokrytii”, Problemy kibernetiki, 37 (1980), 19–56 | MR | Zbl