Summary. This is the first of two papers in which we classify the regular projective polyhedra in P 3 with planar faces. Here, we develop the basic notions; we introduce a new diophantine trigonometric equation, which plays a key role in the classification theorem, relating the combinatorial and geometric parameters of such polyhedra, and conclude with the case in which the polyhedron is an embedded surface.Mathematics Subject Classification (1991). 52B, 52C, 51M20, 51F15.

The regular projective polyhedra with planar faces and skew vertex figures are classified. There is an infinite family of tori, plus 84 special polyhedra. Correspondingly, but not explicitly, the classification of an important family of regular euclidean polyhedra in 4 dimensional space may be derived. New euclidean regular polytopes in all dimensions are defined.Mathematics Subject Classification (1991). 52B, 52C, 51M20, 51F15.

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger's theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

A k-graph, H = (V,E), is tight if for every surjective mapping f : V + { 1,. . . , k } there exists an edge a E f such that f l , is injective. Clearly, 2-graphs are tight if and only if they are connected. Bounds for the minimum number p," of edges in a tight k-graph with n vertices are given. We conjecture that p : = [n(n -2)/31 for every n and prove the equality when 2n + 1 is prime. From the examples, minimal embeddings of complete graphs into surfaces follow.

Foundations for the topic of crystallizations are proposed through the more general concept of colored triangulations. Classic results and techniques of crystallizations are reviewed from this point of view. A new set of combinatorial invariants of manifolds is defined, and related to the fundamental group and other known invariants. A universal group theoretic approach for this theory is introduced.