caa a22 4500 475778200 CHVBK 20180406123637.0 cr unu---uuuuu 170329e20001201xx s 000 0 eng 10.1007/PL00007228 doi (NATIONALLICENCE)springer-10.1007/PL00007228 McGuinness Sean Department of Mathematics, University of Umeå, S-90187 Umeå, Sweden e-mail: sean@abel.math.umu.se, SE aut Colouring Arcwise Connected Sets in the Plane I [Elektronische Daten] [Sean McGuinness] Abstract.: Let ? be the family of finite collections ? where ? is a collection of bounded, arcwise connected sets in ℝ2 which for any S, T∈? where S∩T≠∅, it holds that S∩T is arcwise connected. We investigate the problem of bounding the chromatic number of the intersection graph G of a collection ?∈?.  Assuming G is triangle-free, suppose there exists a closed Jordan curve C⊂ℝ2 such that C intersects all sets of ? and for all S∈?, the following holds: (i) S∩(C∪int (C)) is arcwise connected or S∩int (C)=∅. (ii) S∩(C∪ext (C)) is arcwise connected or S∩ext (C)=∅.  Here int(C) and ext (C) denote the regions in the interior, resp. exterior, of C. Such being the case, we shall show that χ(?) is bounded by a constant independent of ?. Springer-Verlag Tokyo, 2000 https://doi.org/10.1007/PL00007228 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/PL00007228 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- McGuinness Sean Department of Mathematics, University of Umeå, S-90187 Umeå, Sweden e-mail: sean@abel.math.umu.se, SE aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence SWISSBIB 475778200 BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer