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  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  of certain systems of set pairs.
Received April 4, 2000, and in revised form January 4, 2001. Online publication August 28, 2001.
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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