Colouring Arcwise Connected Sets in the Plane II

Let G be the family of nite collections S where S is a collection of closed, bounded, arcwise connected sets in R 2 which for any S; T 2 S where S \ T 6 = ;, it holds that S \ T is arcwise connected. Given S 2 G which is triangle-free, we show that provided S is suu-ciently large there exists a subcollection S 0 S of at most 5 sets with the property that the sets of S surrounded by S 0 induce an intersection graph H where (H) 1 (G), and > 1 does not depend on S. In conjuction with this result, we obtain a new result concerning the so-called L-graph conjecture. We show that if for a triangle-free collection of L-shapes L it holds that for any two intersecting L-shapes the ratio of their horizontal lengths and the ratio of their vertical lengths are bounded above by a constant independent of L, then the chromatic number of the L-graph G(L) is bounded above by a constant depending only on. Any of the notation and concepts not explicitly deened here can be found in 13]. For a collection F of subsets of R n we deene the intersection graph G(F) of F to be the graph whose vertices correspond to sets in F where two vertices are adjacent if and only if there corresponding sets have nonempty intersection. Assuming G is as deened in the abstract , we let G 3 denote the set of all triangle-free collections of G. It is known that graphs can be found which have arbitrarily high chromatic number and girth(see 4]). We shall show that for a collection S 2 G 3 , there exists S 2 S such that the collection of sets of S at distance 2 from S has chromatic number which is at least a certain fraction of (S), this fraction being independent of S. For a collection of sets S from R 2 and S 0 S we let I S (S 0) denote the collection of sets of S which are surrounded by sets of S 0 ; that is, those sets contained in the nite components of R 2 ? S S2S 0 S: When S is implicit, we shall just write I(S 0). For convenience we shall write I(S 1 ; S 2 ; : : : ; S n) instead …