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