Geodesic
Combinatorial Extremum Problems for $2$-Colorings of Hypergraphs
Matematičeskie zametki, Tome 90 (2011) no. 4, pp. 584-598

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We consider a generalization of the Erdős–Hajnal classical combinatorial problem. Let $k$ be a positive integer. It is required to find the value of $m_k(n)$ equal to the minimum number of edges of an $n$-uniform hypergraph that does not admit $2$-colorings of the set of its vertices such that each edge of the hypergraph contains exactly $k$ vertices of each color. In the present paper, we obtain a new asymptotic lower bound for $m_k(n)$, which improves the preceding results in a wide range of values of the parameter $k$. We also consider some other generalizations of this problem.
Keywords: $n$-uniform hypergraph, $2$-coloring, asymptotic lower bound. 
@article{MZM_2011_90_4_a7,
     author = {A. P. Rozovskaya},
     title = {Combinatorial {Extremum} {Problems} for $2${-Colorings} of {Hypergraphs}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {584--598},
     publisher = {mathdoc},
     volume = {90},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_4_a7/}
}
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A. P. Rozovskaya. Combinatorial Extremum Problems for $2$-Colorings of Hypergraphs. Matematičeskie zametki, Tome 90 (2011) no. 4, pp. 584-598. http://geodesic.mathdoc.fr/item/MZM_2011_90_4_a7/