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 Collections: Mathematics