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Abstract: Let be an orthogonal array and let be an -coloring of its ground set . We give a combinatorial identity which relates the number of vectors in with given color patterns under with the cardinalities of the color classes. Several applications of the identity are considered. Among them it is shown that every coloring of an orthogonal array contains a positive proportion of almost rainbow vectors and also of almost monochromatic vectors of every color. [Copyright &y& Elsevier]

Abstract: In this paper, we study homomorphisms of -edge-colored graphs, that is graphs with edges colored with two colors. We consider various graph classes (outerplanar graphs, partial 2-trees, partial -trees, planar graphs) and the problem is to find, for each class, the smallest number of vertices of a -edge-colored graph such that each graph of the considered class admits a homomorphism to . [Copyright &y& Elsevier]

We study vertex colorings of hypergraphs, such that all color sizes differ at most in one (balanced colorings) and each edge contains at least two vertices of the same color (rainbow-free colorings). Given a hypergraph H , the maximum k , such that there is a balanced rainbow-free k -coloring of H is called the balanced upper chromatic number denoted by χ ¯ b ( H ) . Concerning hypergraphs defined by projective spaces, bounds on the balanced upper chromatic number and constructions of rainbow-free colorings are given. For cyclic projective planes of order q we prove that: q 2 + q + 1 6 ≤ χ ¯ b ( Π q ) ≤ q 2 + q + 1 3 . We also give bounds for the balanced upper chromatic numbers of the hypergraphs arising from the n -dimensional finite space PG ( n , q ) . [ABSTRACT FROM AUTHOR]