We improve some upper bounds for minimal dispersion on the cube and torus. Our new ingredient is an improvement of a probabilistic lemma used to obtain upper bounds for dispersion in several previous works. Our new lemma combines a random and non-random choice of points in the cube. This leads to better upper bounds for the minimal dispersion.

Abstract. Suppose a committee consisting of three members is tasked with matching [Formula: see text] candidates to [Formula: see text] different positions. However, all of the committee members disagree on the job placement for every candidate, i.e., every candidate is matched to three different positions according to three committee members. All three committee members are competitive and want to push through as many of their placements as possible. Can they find a compromise which allows each committee member to be responsible for a third of all the candidate placements? In this paper we will consider an asymptotic version of this question and several other variants of a similar problem. As an application we will consider an embedding question—which hypertrees does a large Steiner system always contain?