2024/09/11 by Arun, Srinivas, Dillon, Travis
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2409.07262
A finite family \mathcal F of convex sets is k-intersecting in S ⊆ ℝd if the intersection of every subset of k convex sets in \mathcal F contains a point in S. The Helly number of S is the minimum k, if it exists, such that every k-intersecting family contains a point of S in its intersection. In this paper, we improve bounds on the Helly number of product sets of the form Ad for various sets A ⊆ ℝ, including the ``exponential grid'' A = \αn : n ∈ ℕ\ and sets A⊆ ℤ defined by congruence relations.