2023/01/11 by Ambrus, Gergely, Balko, Martin, Frankl, Nóra +2
#52A35 #52A38 #52C07 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2301.04683
Given a set S ⊆ ℝ2, define the Helly number of S, denoted by H(S), as the smallest positive integer N, if it exists, for which the following statement is true: for any finite family F of convex sets in~ℝ2 such that the intersection of any N or fewer members of~F contains at least one point of S, there is a point of S common to all members of F. We prove that the Helly numbers of exponential lattices \αn \colon n ∈ ℕ0\2 are finite for every α>1 and we determine their exact values in some instances. In particular, we obtain H(\2n \colon n ∈ ℕ0\2)=5, solving a problem posed by Dillon (2021). For real numbers α, β> 1, we also fully characterize exponential lattices L(α,β) = \αn \colon n ∈ ℕ0\ × \βn \colon n ∈ ℕ0\ with finite Helly numbers by showing that H(L(α,β)) is finite if and only if logα(β) is rational.