Generalized Davenport-Schinzel sequences

The extremal functionEx(u, n) (introduced in the theory of Davenport-Schinzel sequences in other notation) denotes for a fixed finite alternating sequenceu=ababa... the maximum length of a finite sequencev overn symbols with no immediate repetition which does not containu. Here (following the idea of J. Nešetřil) we generalize this concept for arbitrary sequenceu. We summarize the already known properties ofEx(u, n) and we present also two new theorems which give good upper bounds onEx(ui,n). We use these theorems to describe a wide class of sequencesu (“linear sequences”) for whichEx(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 aboutEx(u, n).

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