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The goal of this paper is to give a new, abstract approach tocover-decomposition and polychromatic colorings using hypergraphs on orderedvertex sets. We introduce an abstract version of a framework by Smorodinsky andYuditsky, used for polychromatic coloring halfplanes, and apply it to so-calledABA-free hypergraphs, which are a generalization of interval graphs. Using ourmethods, we prove that (2k-1)-uniform ABA-free hypergraphs have a polychromatick-coloring, a problem posed by the second author. We also prove the same forhypergraphs defined on a point set by pseudohalfplanes. These results are bestpossible. We could only prove slightly weaker results for dual hypergraphsdefined by pseudohalfplanes, and for hypergraphs defined by pseudohemispheres.We also introduce another new notion that seems to be important forinvestigating polychromatic colorings and epsilon-nets, shallow hitting sets.We show that all the above hypergraphs have shallow hitting sets, if theirhyperedges are containment-free.

An abstract approach to polychromatic coloring: shallow hitting sets in ABA-free hypergraphs and pseudohalfplanes