Zoltan Furedi

Z. Furedi: Applications of Turan theory in combinatorial geometry

We overview a few applications of Turan graph theory in extremal combinatorial geometry. An example of results (and questions): we show that any n point set P (in general position) in the three dimensional space contains a pair x,y\in P such that the open ball B(x,y) with diameter [xy] contains n/30 additional members of P.
Transparencies of a lecture presented on
Bernoulli Conference, Lausanne, Aug 31, 2010
Geometry Seminar, University of Illinois at Urbana-Champaign, April 12, 2011

page 1 Title page
page 2 Summary
page 3 The Erdos-Stone-Simonovits thm
page 4 Kovari-T Sos-Turan: the bipartite case
page 5 Distribution of distances
page 6 Sums of vectors (Katona)
page 7 Vectors with small pairwise sums
page 8 Cells and lines
page 9 Antipodal pairs
page 10 Nearly equal distances
page 11 Popular distances in R3
page 12 Repeated angles
page 13 Unit distances in R2
page 14 Unit distances in R3
page 15 Unit distances in Rd
page 16 All boxes are empty
page 17 The number of empty boxes in R2
page 18 The number of empty boxes in Rd
page 19 The number of points in axis parallel boxes
page 20 Same in Rd
page 21 The problem of number of points in a sphere
page 22 The case of R2
page 23 The number of points in a Thales sphere
page 24 The proof with double counting
page 25 The proof with hypergraph Turan thm