Piercing Random Boxes Linh V. Tran Department of Mathematics, Rutgers, Piscataway, New Jersey 08854; e-mail: linhtran@math.rutgers.edu Received 18 February 2009; accepted 10 August 2009 Published online 11 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/rsa.20321 ABSTRACT: Consider a set of n random axis parallel boxes in the unit hypercube in R d , where d is fixed and n tends to infinity. We show that the minimum number of points one needs to pierce all these boxes is, with high probability, at least  d ( √ n(log n) d/2−1 ) and at most O d ( √ n(log n) d/2−1 log log n). © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 365–380, 2011 Keywords: random boxes; random hypergraph; discrepancy 1. INTRODUCTION Let U d be the unit hypercube in R d , where d is a constant: U d :=[0, 1] d . We generate a d -dimensional random box by taking the product of d independent random sub-intervals of [0, 1], where each random interval is determined by two random (end-) points, each chosen independently with respect to the uniform measure on [0, 1]. Consider a family of n (independently) random boxes. Our investigation is motivated by the following basic questions: What is the size of the largest sub-family of pairwise disjoint boxes? What is the minimum number of points one needs to pierce all the boxes? Let us denote these quantities by ν d (n) and τ d (n), respectively. These quantities, usually referred to as the matching and covering (piercing) numbers, are of fundamental interest and have been studied for a large variety of hypergraphs, deterministic and random alike. It is useful to notice that the piercing number is at least the matching number, by definition. About 10 years ago, Coffman et al. studied ν d (n) [1]. They showed Theorem 1.1. For d = 2, ν 2 (n) = ( √ n). For d ≥ 3 ( √ n) ≤ ν d (n) ≤ O   n log d−1 n  . (1) Correspondence to: L. V. Tran © 2010 Wiley Periodicals, Inc. 365