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6. INTERVALS The purpose of this last section is to obtain a criterion for the existence of transversals to a set of intervals in Rn. By an interval we mean any connected subset of a line, so that they may also be open, rays or complete lines. A line being transversal to a set of intervals means that it does intersect all of them. 1 to intervals, requires the Hadwiger hypothesis of partial transversals compatible with a given linear order. Because 14 AROCHA, BRACHO AND MONTEJANO we have been working in projective space, where the general problem really lies, one other natural Hadwiger-type hypothesis to consider is compatibility with a given cyclic order.
O l r _ f c ; o n _ f c + i , o n _ f c + 2 , . . ,o n }, ij G { 1 , 2 , . . ,r — k belongs exactly to two classes containing the chosen ktuple ffki an(l therefore these (r — fc)-tuple belongs to exactly one such pair of classes. So, it must be true that (2) > ("l£)- Hence, zAi/-2(™~£) > 0. From this we get the desired inequality. : On a certain class of incidence structures. Prace a stiidie Vysokej skoly dopravnej v Ziline 2 (1979), 97-106. : On a connection between unit circles and horocycles determined by n points.
Geometry (ED. -R. Sack and J. Urrutia), pp. 633-701. V. North-Holland, Amsterdam, 2000. [5] C. H. Papadimitriou, and M. Yannakakis, Shortest paths without a map, Jour. Proc. 16th Internat. Colloq. on Automata, Languages, and Programming, Lecture Notes in Computer Science 372, 610-620, SpringerVerlag (1989). edu ''Donald R. edu ''Andras Bezdek'' A SHORT SURVEY OF (r,g)-STRUCTURES VOJTECH BALINT1 Department of Mathematics, University of Zilina, 010 26 Zilina, Slovakia. ABSTRACT. This contribution gives a short survey of results on the combinatorial (r, g)-structures as the generalization of many geometrical structures.