Weak ε-nets and interval chains

We construct weak ε-nets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1/<i>r</i>-nets of size <i>O(rα(r))</i>, where <i>α(r)</i> denotes the inverse Ackermann function. For point sets along the moment curve in ℝ<i>^d</i> we construct weak 1/<i>r</i>-nets of size <i>r</i> · 2^{poly<i>(α(r))</i>}, where the degree of the polynomial in the exponent depends (quadratically) on <i>d.</i> Our constructions result from a reduction to a new problem, which we call <i>stabbing interval chains with j-tuples.</i> Given the range of integers <i>N</i> = [1,<i>n</i>], an interval chain of length <i>k</i> is a sequence of <i>k</i> consecutive, disjoint, nonempty intervals contained in <i>N.</i> A <i>j</i>-tuple &pmacr; is said to <i>stab</i> an interval chain <i>C = I₁ … I_k</i> if each <i>p_i</i> falls on a different interval of <i>C.</i> The problem is to construct a small-size family Z of <i>j</i>-tuples that stabs all <i>k</i>-interval chains in <i>N.</i> Let <i>z_k^(j)(n)</i> denote the minimum size of such a family Z. We derive almost-tight upper and lower bounds for <i>z_k^(j)(n)</i> for every fixed <i>j;</i> our bounds involve functions <i>α_m(n)</i> of the inverse Ackermann hierarchy. Specifically, we show that for <i>j</i> = 3 we have <i>z_k⁽³⁾(n)</i> = Θ for all <i>k</i> ≥ 6. For each <i>j</i> ≥ 4 we construct a pair of functions <i>P'_j(m), Q'_j(m)</i>, almost equal asymptotically, such that <i>z(j)_{P'_j(m)}(n)=O(nα_m(n))</i> and <i>z(j)_{Q'_j(m)}(n)=Ω(nα_m(n))</i>.