Some results in combinatorial geometry

Some Results in Geometric Topology and Geometry

Some Recent Results in Cr Geometry

CR geometry originated from the study of real hypersurfaces in complex manifolds. In their foundational work Chern and Moser [CM] developed local invariants for CR manifolds. Shortly afterward Tanaka [T] and Webster [W2] introduced a canonical connection associated to a given pseudohermitian structure (i.e. a contact form), so pseudohermitian geometry was born. Today CR geometry or pseudohermit...

Some Old and New Problems in Combinatorial Geometry

Usually we the number the points Let x l ,x , . . .,xn be n distinct points in a metric space . will restrict ourselves to the plane . Denote by D(x , . . .,x ) of distinct distances determined by x , . . .,x . Assume lthat n are in r-dimensional space . Denote byl n f (n)= min D(x, . . .,x), r xl, . . .,x l n n I conjectured more than 40 years ago that f (n)> c n/(log n) 2 . The lattice points...

Ideal versions of some combinatorial results

We characterize ideals of subsets of natural numbers for which some versions of Ramsey’s theorem and Schur’s theorem hold. These are similar to generalizations shown by Frankl, Graham and Rödl in [6]. We show that, in some sense, Ramsey’s theorem, theorem of Bolzano-Weierstrass and “monotone subsequence” theorem characterize the same class of ideals.

Part 1 . Discrete Problems Problems and Results in Combinatorial Geometry °

I have published during my very long life many papers with similar titles . In this note, I report on progress made on some of the old problems and discuss also some new ones, and I repeat some old problems which seem to have been forgotten by everybody (sometimes including myself) . First, a very incomplete list is given of some papers dealing with these and related questions . I have publishe...