Coloring [Formula: see text]-Embeddable [Formula: see text]-Uniform Hypergraphs

This paper extends the scenario of the Four Color Theorem in the following way. Let [Formula: see text] be the set of all [Formula: see text]-uniform hypergraphs that can be (linearly) embedded into [Formula: see text]. We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in [Formula: see text]. For example, we can prove that for [Formula: see text] there are hypergraphs in [Formula: see text] on [Formula: see text] vertices whose chromatic number is [Formula: see text], whereas the chromatic number for [Formula: see text]-vertex hypergraphs in [Formula: see text] is bounded by [Formula: see text] for [Formula: see text].