 
Summary: Weak nets and interval chains
Noga Alon, Haim Kaplan, Gabriel Nivasch1
,
Micha Sharir, Shakhar Smorodinsky2
July 17, 2007
Abstract
We construct weak nets of almost linear size for certain types of point sets.
Specifically, for planar point sets in convex position we construct weak 1
r nets of
size O(r(r)), where (r) denotes the inverse Ackermann function. For point sets
along the moment curve in Rd
we construct weak 1
r nets of size r · 2poly((r))
, where
the degree of the polynomial in the exponent depends (quadratically) on d.
Our constructions result from a reduction to a new problem, which we call
stabbing interval chains with jtuples. Given the range of integers N = [1, n], an
interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals
contained in N. A jtuple p = (p1, . . . , pj) is said to stab an interval chain C =
I1 · · · Ik if each pi falls on a different interval of C. The problem is to construct a
