Abstract
Any finite set X⊂ℝd colored with \(\lfloor\frac {d+3}{2}\rfloor\) colors, contains a rainbow subset Y⊂X, such that any ball that contains Y contains a positive fraction of the points of X. The bound on the number of colors is tight.
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This work was done while the author was visiting the Mathematical Sciences Research Institute, Berkeley. The author was supported by the UCL Graduate School Research Grant, #1.03/04.RP.EMB.
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Prodromou, M.N. A Combinatorial Property of Points and Balls, a Colored Version. Discrete Comput Geom 38, 641–650 (2007). https://doi.org/10.1007/s00454-007-9003-4
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DOI: https://doi.org/10.1007/s00454-007-9003-4