Discrete & Computational Geometry

, Volume 18, Issue 2, pp 195–203

Transversals of d-Intervals

  • T. Kaiser

DOI: 10.1007/PL00009315

Cite this article as:
Kaiser, T. Discrete Comput Geom (1997) 18: 195. doi:10.1007/PL00009315


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.

Copyright information

© 1997 Springer-Verlag New York Inc.

Authors and Affiliations

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