Improved lower bounds on the length of Davenport-Schinzel sequences

Abstract

We derive lower bounds on the maximal lengthλ s(n) of (n, s) Davenport Schinzel sequences. These bounds have the form λ2s=1(n)=Ω(nαs(n)), whereα(n) is the extremely slowly growing functional inverse of the Ackermann function. These bounds extend the nonlinear lower boundλ 3 (n)=Ω(nα(n)) due to Hart and Sharir [5], and are obtained by an inductive construction based upon the construction given in [5].

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Work on this paper has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation.

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Sharir, M. Improved lower bounds on the length of Davenport-Schinzel sequences. Combinatorica 8, 117–124 (1988). https://doi.org/10.1007/BF02122559

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AMS subject classification (1980)

  • 05 A 99
  • 05 C 35
  • 68 B 15