We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions.
@InProceedings{keszegh_et_al:LIPIcs.SoCG.2017.47, author = {Keszegh, Bal\'{a}zs and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r}, title = {{Proper Coloring of Geometric Hypergraphs}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {47:1--47:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.47}, URN = {urn:nbn:de:0030-drops-71926}, doi = {10.4230/LIPIcs.SoCG.2017.47}, annote = {Keywords: discrete geometry, decomposition of multiple coverings, geometric hypergraph coloring} }
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