… the weak ϵ-nets. At the end of Section 3, we address the issue of constructing our weak ϵ-nets …
In Section 4, we derive our almost-matching lower bounds for stabbing interval chains, as …

… intervals on the xl-axis connecting pairs of points in S1, and construct an interval tree over
these intervals… half of the intervals lie fully to the left of a and at most half lie fully to its right. The …

… As Alon, Kalai, Matoušek, and Meshulam noted in 2001: “Finding the correct estimates
for weak ϵ-nets is, in our opinion, one of the truly important open problems in combinatorial …

… approximations and ϵ-nets—that … weak ϵ-nets induced by convex sets and ϵ-nets for the
primal set system induced by intersections of half-spaces, though the resulting size of the weak …

… We call an interval Hx,j bad if its size is at least 4r, oth… exist weak ε-nets of size O(1). However,
we have been unable to decide whether the analogue of Theorem 4 holds for weak ε-nets …

… As the primary focus of this study is on the combinatorial aspects of weak ϵ-nets, we did
not seek to optimize the construction cost of our net Q. A straightforward implementation of the …

… is under what conditions do weak nets exist. This question for … number characterizes the
existence of weak nets in a pretty … intervals in the well-ordering of X (a set \(I\subseteq X\) is an …

… 2 Intervals. Consider the set X to be the real line, and consider R to be the set of all intervals
… Proof: As a warm-up exercise, we prove a somewhat weaker bound here of O((δ + δ )log(δ + …

… superlinear lower bound for weak ε-nets in a fixed dimension. … We also consider weak ε-nets
for the diagonal of our stretched … We now recall the problem of stabbing interval chains and …

… While an ɛdash-net is a weaker notion of approximation, its … , for each interval chain I with
at least k intervals, there … that, for each interval chain I in [t], if αt is the number of intervals in I, …