J. Geom. 101 (2011), 83–97 c  2011 Springer Basel AG 0047-2468/11/010083-15 published online October 19, 2011 DOI 10.1007/s00022-011-0082-2 Journal of Geometry Maximum number of colors: C-coloring and related problems Csilla Bujt´as and Zsolt Tuza Abstract. We discuss problems and results on the maximum number of colors in combinatorial structures under the assumption that no totally multicolored sets of a speciﬁed type occur. Mathematics Subject Classiﬁcation (2010). 05C15, 05C65, 05C35, 05B05. Keywords. Hypergraph coloring, C-coloring, partition crossing, C-perfect hypergraph, Steiner system, ﬁnite projective plane, 3-consecutive coloring. 1. The general concept of C-coloring In this paper we survey problems and results concerning a kind of partitions of combinatorial structures, called C-coloring. In general, by C-coloring we mean that each of the speciﬁed objects in the structure has two elements with a c ommon color; that is, belonging to the same partition class. In this context the “speciﬁed objects” can mean, for example, subgraphs of given type in a graph, faces of a polyhedron, blocks of a Steiner system, or quite generally those members of a set system which have cardinality at least two. Assigning the same color to all elements always is a C-coloring; that is, one color is feasible in any case. Hence, the most essential parameter of a system with respect to C-coloring is the largest possible number of colors, which is called upper chromatic number and is denoted by χ. As a comparison, in the classical proper coloring one requires at least two ele- ments of distinct colors in every speciﬁed object. Concerning proper colorings the maximum number of colors trivially is achieved by assigning mutually dif- ferent colors to all elements; therefore one traditionally asks about the smallest This work was supported in part by the Hungarian Scientiﬁc Research Fund, OTKA Grant 81493.