Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Lower Bounds on the Transversal Numbers of d-Intervals

  • 114 Accesses

  • 8 Citations

Abstract

Let l 1 ,l 2 ,\ldots,l d be disjoint parallel lines in the plane. A d-interval is a subset of their union that intersects each l i in a closed interval. Kaiser [4] showed that any system of d -intervals containing no subsystem of k+1 pairwise disjoint d -intervals can be pierced by at most (d 2 -d)k points. We show that this bound is close to being optimal, by proving a lower bound of const(d 2 /log 2 d)k. The construction involves an extension of a construction due to Sgall [8] of certain systems of set pairs.

Author information

Additional information

Received April 4, 2000, and in revised form January 4, 2001. Online publication August 28, 2001.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Matoušek, J. Lower Bounds on the Transversal Numbers of d-Intervals. Discrete Comput Geom 26, 283–287 (2001). https://doi.org/10.1007/s00454-001-0037-8

Download citation