ORIGINS OF NONLINEARITY IN DAVENPORT-SCHINZEL SEQUENCES

• Published in: SIAM journal on discrete mathematics Vol. 25; no. 1-2; pp. 211 - 233
• Main Author: PETTIE, Seth
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2011
• Subjects: Algebra; Alphabets; Applied mathematics; Combinatorics; Combinatorics. Ordered structures; Construction; Exact sciences and technology; Graphs; Mathematical analysis; Mathematics; Nonlinear equations; Nonlinearity; Origins; Sciences and techniques of general use; Studies; Letter; Maximum; Area; 05D99; Complete graph; 68R05; infinite antichains; Discrete mathematics; Nonlinearity; Inverse; forbidden sequences; generalized Davenport-Schinzel sequences
• ISSN: 0895-4801; 1095-7146
• DOI: 10.1137/080735862

A generalized Davenport-Schinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Ex(σ, n) be the maximum length of a sequence over an n-letter alphabet excluding subsequences isomorphic to σ. It has been proved that for every σ, Ex(σ, n) is either linear or very close to linear. In particular it is O(...), where α is the inverse-Ackermann function and O(1) depends on σ. In much the same way that the complete graphs K5 and K3,3 represent the minimal causes of nonplanarity, there must exist a set ΦNonlin of minimal nonlinear forbidden subsequences. Very little is known about the size or membership of ΦNonlin. In this paper we construct an infinite antichain of nonlinear forbidden subsequences which, we argue, strongly supports the conjecture that ΦNonlin is itself infinite. Perhaps the most novel contribution of this paper is a succinct, humanly readable code for expressing the structure of forbidden subsequences. [PUBLICATION ABSTRACT]