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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(More)
A geometric graph is a graph G = (V;E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . It is known that, for any xed k, any geometric graph G on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. In this paper we give a(More)
We prove a fractional version of the Erdős–Szekeres theorem: for any k there is a constant ck > 0 such that any sufficiently large finite set X ⊂ R2 contains k subsets Y1, . . . ,Yk , each of size ≥ ck |X |, such that every set {y1, . . . , yk} with yi ∈ Yi is in convex position. The main tool is a lemma stating that any finite set X ⊂ Rd contains “large”(More)
A geometric graph is a graph G = (V;E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . Two edges of a geometric graph are said to be parallel , if they are opposite sides of a convex quadrilateral. In this paper we show that, for any xed k 3, any(More)
Let C be a system (finite or infinite) of centrally symmetric convex bodies in IR with disjoint interiors; we call such a C a packing . For a real number ε > 0 and for C ∈ C, we let C denote C enlarged by the factor 1+ ε from its center, that is, C = (1+ ε)(C − c) + c, where c stands for the center of symmetry C. Let us call the closure of the set C \C the(More)
A geonuh=ic graph. is a graph drawn in the plane so that bhe vertices are represented by points in general posit.ion, the edges are represented by straight line segments connect,ing the corresponding points. Improving a result of Path and TM&k, we show that a geometric graph on n vertices with no X: + 1 pairwise dis-ioint edges has at most b3(n+ 1) edges.(More)
A <italic>halving hyperplane</italic> of a set <italic>S</italic> of <italic>n</italic> points in <bold>R</bold><supscrpt><italic>d</italic></supscrpt> contains <italic>d</italic> affinely independent points of <italic>S</italic> so that equally many of the points off the hyperplane lie in each of the two half-spaces. We prove bounds on the number of(More)