Discrete & Computational Geometry

, Volume 18, Issue 2, pp 195-203

First online:

Transversals of d-Intervals

  • T. KaiserAffiliated withDepartment of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic kaiser@kam.ms.mff.cuni.cz

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We present a method which reduces a family of problems in combinatorial geometry (concerning multiple intervals) to purely combinatorial questions about hypergraphs. The main tool is the Borsuk—Ulam theorem together with one of its extensions.

For a positive integer d, a homogeneous d-interval is a union of at most d closed intervals on a fixed line . Let \({\cal H}\) be a system of homogeneous d-intervals such that no k + 1 of its members are pairwise disjoint. It has been known that its transversal number \(\tau ({\cal H})\) can then be bounded in terms of k and d. Tardos [9] proved that for d = 2, one has \(\tau ({\cal H}) \leq 8k\) . In particular, the bound is linear in k. We show that the latter holds for any d, and prove the tight bound \(\tau ({\cal H}) \leq 3k\) for d = 2.

We obtain similar results in the case of nonhomogeneous d-intervals whose definition appears below.