We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.
@InProceedings{damasdi_et_al:LIPIcs.SoCG.2022.32, author = {Dam\'{a}sdi, G\'{a}bor and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r}, title = {{Three-Chromatic Geometric Hypergraphs}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {32:1--32:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.32}, URN = {urn:nbn:de:0030-drops-160401}, doi = {10.4230/LIPIcs.SoCG.2022.32}, annote = {Keywords: Discrete geometry, Geometric hypergraph coloring, Decomposition of multiple coverings} }
Feedback for Dagstuhl Publishing