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**1 - 5**of**5**### Density version of the Ramsey problem and the directed Ramsey problem

"... We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on n vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges |ERB| is given. The aim is to find the maximal size f of a mono ..."

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We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on n vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges |ERB| is given. The aim is to find the maximal size f of a monochromatic clique which is guaranteed by such a coloring. Analogously, in the second problem we consider semicomplete digraph on n vertices such that the number of bi-oriented edges |Ebi| is given. The aim is to bound the size F of the maximal transitive subtournament that is guaranteed by such a digraph. Applying probabilistic and analytic tools and constructive methods we show that if |ERB | = |Ebi | = p

### Independence and Matchings in σ-hypergraphs

"... Let σ be a partition of the positive integer r. A σ-hypergraph H = H(n, r, q|σ) is an r-uniform hypergraph on nq vertices which are parti-tioned into n classes V1, V2,..., Vn each containing q vertices. An r-subset K of vertices is an edge of the hypergraph if the partition of r formed by the non-ze ..."

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Let σ be a partition of the positive integer r. A σ-hypergraph H = H(n, r, q|σ) is an r-uniform hypergraph on nq vertices which are parti-tioned into n classes V1, V2,..., Vn each containing q vertices. An r-subset K of vertices is an edge of the hypergraph if the partition of r formed by the non-zero cardinalities |K ∩ Vi|, 1 ≤ i ≤ n, is σ. In earlier works we have considered colourings of the vertices of H which are constrained such that any edge has at least α and at most β vertices of the same colour, and we have shown that interesting results can be obtained by varying α, β and the parameters of H appropriately. In this paper we continue to investigate the versatility of σ-hypergraphs by considering two classical problems: independence and matchings. We first demonstrate an interesting link between the constrained colour-ings described above and the k-independence number of a hypergraph, that is, the largest cardinality of a subset of vertices of a hypergraph not con-taining k+1 vertices in the same edge. We also give an exact computation of the k-independence number of the σ-hypergraph H. We then present results on maximum, and sometimes perfect, matchings in H. These re-sults often depend on divisibility relations between the parameters of H and on the highest common factor of the parts of σ. 1