What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely pseudo-disks, with a non-empty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way and our proof that they are 3-chromatic is also combinatorial.
|Number of pages||8|
|Journal||Acta Mathematica Universitatis Comenianae|
|State||Published - 2 Sep 2019|
Bibliographical noteFunding Information:
Received June 2, 2019. 2010 Mathematics Subject Classification. Primary 05C15. Research by the first author was partially supported by ERC AdG Disconv and MTA EU10/2016-11001. Research by the second author was supported by the National Research, Development and Innovation Office – NKFIH under the grant K 116769. Research by the second and third authors was supported by the Lendület program of the Hungarian Academy of Sciences (MTA), under grant number LP2017-19/2017.
© 2019, Univerzita Komenskeho. All rights reserved.
ASJC Scopus subject areas
- Mathematics (all)