Shira Zerbib

Fractional covers and matchings in families of weighted d-intervals

A d-interval is a union of at most d disjoint closed intervals on a fixed line. Tardos [14] and the second author [11] used topological tools to bound the transversal number τ of a family H of d-intervals in terms of d and the matching number ν of H. We investigate the weighted and fractional versions of this problem and prove upper bounds that are tight up to constant factors. We apply both a topological method and an approach of Alon [1]. For the use of the latter, we prove a weighted version of Turán’s theorem. We also provide proofs of the upper bounds of [11] that are more direct than the original proofs.

Edge-Covers in d-Interval Hypergraphs

A d-interval hypergraph has d disjoint copies of the unit interval as its vertex set, and each edge is the union of d subintervals, one on each copy. Extending a classical result of Gallai on the case

The number of distinct distances from a vertex of a convex polygon

d=1

Piercing Axis-Parallel Boxes

d=1, Tardos and Kaiser used topological tools to bound the ratio between the transversal number and the matching number in such hypergraphs. We take a dual point of view, and bound the edge-covering number (namely the minimal number of edges covering the entire vertex set) in terms of a parameter expressing independence of systems of partitions of the d unit intervals. The main tool we use is an extension of the KKM theorem to products of simplices, due to Peleg.

The

Let L be a set of n lines in the real projective plane in general position. We show that there exists a vertex v∈𝒜(L) such that v is positioned in a face of size at most 5 in the arrangement obtained by removing the two lines passing through v.