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The extremal function Ex(u,n) (introduced in the theory of Davenport-Schinzel 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. Ne~et~il) 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 u i 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).

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

We prove a fractional version of the Erdős-Szekeres theorem: for any k there is a constant c k > 0 such that any sufficiently large finite set X ⊂ R 2 contains k subsets Y 1 , . . . , Y k , each of size ≥ c k |X |, such that every set {y 1 , . . . , y k } with y i ∈ Y i is in convex position. The main tool is a lemma stating that any finite set X ⊂ R d contains "large" subsets Y 1 , . . . , Y k such that all sets {y 1 , . . . , y k } with y i ∈ Y i have the same geometric (order) type. We also prove several related results (e.g., the positive fraction Radon theorem, the positive fraction Tverberg theorem).

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