Sources of Superlinearity in Davenport-Schinzel Sequences

Abstract

A {em generalized} Davenport-Schinzel sequence is one over a finite alphabet
that contains no subsequences isomorphic to a fixed {em forbidden subsequence}.
One of the fundamental problems in this area is bounding (asymptotically) 
the maximum length of such sequences.
Following Klazar, let $Ex(sigma,n)$ be the maximum length of a sequence over an alphabet
of size $n$ avoiding subsequences isomorphic to $sigma$.
It has been proved that for every $sigma$,
$Ex(sigma,n)$ is either linear or very close to linear; in particular it is 
$O(n2^{alpha(n)^{O(1)}})$, where $alpha$ is the inverse-Ackermann function and $O(1)$
depends on $sigma$.  However, very little is known about the properties
of $sigma$ that induce superlinearity of $Ex(sigma,n)$.

In this paper we exhibit an infinite family of independent superlinear forbidden subsequences.
To be specific, we show that there are 17 {em prototypical} superlinear forbidden 
subsequences, some of which can be made arbitrarily long 
through a simple padding operation.
Perhaps the most novel part of our constructions is a new succinct code for
representing superlinear forbidden subsequences.