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Almost linear upper bounds on the length of general Davenport-Schinzel sequences. (English) Zbl 0636.05004

\((n,s)\)-Davenport-Schinzel sequences are sequences that are composed of \(n\) symbols, and are such that (i) no two adjacent elements are equal, and (ii) there does not exist a (not necessarily contiguous) alternating subsequence of length \(s+2\) of the form \(a...b...a...b...\) for any two distinct symbols \(a\) and \(b\). Davenport-Schinzel sequences arise in the computation and analysis of the lower or upper envelope of a set of functions, and are thus a powerful and versatile tool for a variety of problems in combinatorial and computational geometry.
This paper establishes almost linear upper bounds on the maximum length \(\lambda_s(n)\) of \((n,s)\) Davenport-Schinzel sequences. These bounds are of the form \[ O(n\alpha (n)^{O(\alpha (n)^{s-3})}), \] and they generalize and extend the tight bound \(\Theta(n\alpha(n))\) obtained by S. Hart and the author [ibid. 6, 151–177 (1986; Zbl 0636.05003), see the preceding review] for the special case \(s=3\) \((\alpha(n)\) is the extremely slowly growing functional inverse of Ackermann’s function), and also improve the upper bound \(O(n \log^*n)\) due to E. Szemerédi [Acta Arith. 25, 213–224 (1974; Zbl 0291.05003)].
The results of this paper have later been slightly improved by P. K. Agarwal, the author and P. Shor [J. Comb. Theory, Ser. A 52, No. 2, 228–274 (1989; Zbl 0697.05003)], who have shown that \[ \lambda_{2s}(n) = O(n\cdot 2^{O(\alpha(n)^{s-1})}),\quad\text{for } s\geq 2, \] \[ \lambda_{2s+1}(n) = O(n\cdot \alpha(n)^{O(\alpha(n)^{s-1})}),\quad \text{for } s\geq 2. \] Moreover, these bounds are almost tight for even \(s\), because \[ \lambda_{2s}(n) = \Omega (n\cdot 2^{\Omega (\alpha (n)^{s- 1})}),\quad\text{for } s\geq 2. \]
Reviewer: M. Sharir

MSC:

05A05 Permutations, words, matrices
05A15 Exact enumeration problems, generating functions
11B83 Special sequences and polynomials
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