Generalized Davenport-Schinzel Sequences

The extremal function Ex(u, n) (introduced in the theory of DavenportSchinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa . . . the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following the idea of J. Nešetřil) we generalize this concept for arbitrary sequence u. We summarize the already known properties of Ex(u, n) and we present also two new theorems which give good upper bounds on Ex(u, n) for u consisting of (two) smaller subsequences ui provided we have good upper bounds on Ex(ui, n). We use these theorems to describe a wide class of sequences u (”linear sequences”) for which Ex(u, n) = O(n). Both theorems are used for obtaining new superlinear upper bounds as well. We partially characterize linear sequences over three symbols. We also present several problems about Ex(u, n).