Chaya Keller

Improved bounds on the Hadwiger–Debrunner numbers

AbstractLet HDd(p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in Rd which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger–Debrunner conjecture, Alon and Kleitman proved that HDd(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that

Blockers for Noncrossing Spanning Trees in Complete Geometric Graphs

H{D_d}(p,d + 1)is\tilde O({p^{{d^2} + d}})

Reconstruction of the path graph

HDd(p,d+1)isO˜(pd2+d) .We present several improved bounds: (i) For any

On the smallest sets blocking simple perfect matchings in a convex geometric graph

q \geqslant d + 1,H{D_d}(p,d) = \tilde O({p^{d(\frac{{q - 1}}{{q - d}})}})

Characterization of Co-blockers for Simple Perfect Matchings in a Convex Geometric Graph

q≥d+1,HDd(p,d)=O˜(pd(q−1q−d)) . (ii) For q ≥ log p,