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We present several new results regarding λ s (n), the maximum length of a Davenport-Schinzel sequence of order s on n distinct symbols.First, we prove that λ s (n) ≤ n · 2 (1/t!)α(n) t +O(α(n) t−1 ) for s ≥ 4 even, and λ s (n) ≤ n · 2 (1/t!)α(n) t log 2 α(n)+O(α(n) t ) for s ≥ 3 odd, where t = (s − 2)/2 , and α(n) denotes the inverse Ackermann function. The previous upper bounds, by Agarwal, Sharir, and Shor (1989), had a leading coefficient of 1 instead of 1/t! in the exponent. The bounds for even s are now tight up to lower-order terms in the exponent. These new bounds result from a small improvement on the technique of Agarwal et al.More importantly, we also present a new technique for deriving upper bounds for λ s (n). This new technique is very similar to the one we applied to the problem of stabbing interval chains (Alon et al., 2008). With this new technique we: (1) re-derive the upper bound of λ 3 (n) ≤ 2nα(n) + O n α(n) (first shown by Klazar, 1999); (2) re-derive our own new upper bounds for general s; and (3) obtain improved upper bounds for the generalized Davenport-Schinzel sequences considered by Adamec, Klazar, and Valtr (1992).Regarding lower bounds, we show that λ 3 (n) ≥ 2nα(n) − O(n) (the previous lower bound (Sharir and Agarwal, 1995) had a coefficient of 1 2 ), so the coefficient 2 is tight. We also present a simpler variant of the construction of Agarwal, Sharir, and Shor that achieves the known lower bounds of λ s (n) ≥ n · 2 (1/t!)α(n) t −O(α(n) t−1 ) for s ≥ 4 even.

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(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in R d we construct weak 1 r -nets of size r · 2 poly(α(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 j-tuples. 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 j-tuple p = (p 1 , . . . , p j ) is said to stab an interval chain C = I 1 · · · I k if each p i falls on a different interval of C. The problem is to construct a small-size family Z of j-tuples that stabs all k-interval chains in N .Let z (j) k (n) denote the minimum size of such a family Z. We derive almosttight upper and lower bounds for z (j) k (n) for every fixed j; our bounds involve functions α m (n) of the inverse Ackermann hierarchy. Specifically, we show that for j = 3 we have z (3) k (n) = Θ nα k/2 (n) for all k ≥ 6. For each j ≥ 4 we derive a pair of functions P j (m), Q j (m), almost equal asymptotically, such that z (j) P j (m) (n) = O(nα m (n)) and z (j) Q j (m) (n) = Ω(nα m (n)).

The following result was proved by Bárány in 1982: For every d ≥ 1, there exists c d > 0 such that for every n-pointWe investigate the largest possible value of c d . It was known that c d ≤ 1/ (2 d (d + 1)!) (this estimate actually holds for every point set S). We construct sets showing that c d ≤ (d + 1) −(d+1) , and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, isin his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than γ d n d+1 + O(n d ) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved.We also prove that for every n-point set S ⊂ R d , there exists a (d − 2)-flat that stabs at least c d,d−2 n 3 − O(n 2 ) of the triangles spanned by S, with c d,d−2 ≥ 1 24 (1 − 1/(2d − 1) 2 ). To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution Work by B. Bukh was partially carried out at Tel Aviv University. 322Discrete Comput Geom (2010) 43: [321][322][323][324][325][326][327][328][329][330][331][332][333][334][335][336][337][338] in R d can be divided into 4d − 2 equal parts by 2d − 1 hyperplanes intersecting in a common (d − 2)-flat.

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