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Piercing d-intervals A (homogeneous) d-interval is a union of d closed intervals in the line. Using topological meth-
 

Summary: Piercing d-intervals
Noga Alon
Abstract
A (homogeneous) d-interval is a union of d closed intervals in the line. Using topological meth-
ods, Tardos and Kaiser proved that for any finite collection of d-intervals that contains no k + 1
pairwise disjoint members, there is a set of O(d2
k) points that intersects each member of the
collection. Here we give a short, elementary proof of this result.
A (homogeneous) d-interval is a union of d closed intervals in the line. Let H be a finite collection
of d-intervals. The transversal number (H) of H is the minimum number of points that intersect every
member of H. The matching number (H) of H is the maximum number of pairwise disjoint members
of H. GyŽarfŽas and Lehel [3] proved that O(d!) and Kaiser [4] proved that O(d2). His
proof is topological, applies the Borsuk-Ulam theorem and extends and simplifies a result of Tardos
[5]. Here we give a very short, elementary proof of a similar estimate, using the method of [2].
Theorem 1 Let H be a finite family of d-intervals containing no k + 1 pairwise disjoint members.
Then (H) 2d2k.
Proof. Let H be any family of d-intervals obtained from H by possibly duplicating some of its
members, and let n denote the cardinality of H . Note that H contains no k + 1 pairwise disjoint
members. Therefore, by TurŽan's Theorem, there are at least n(n - k)/(2k) unordered intersecting
pairs of members of H . Each such intersecting pair supplies at least 2 ordered pairs (p, I), where p is

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics