Some combinatorial problems in the plane

Some Combinatorial Problems in the Plane

1 Let there be given n points in the plane. Denote by t i the number of lines which contain exactly i of the points (2 < i < n). The properties of the set {ti } have been studied a great deal. For example, there is the classical result of Gallai and Sylvester : Assume to = 0 (i .e ., the points are not all on one line) ; then t2 > 0. For the history of this problem see, e.g ., Motzkin [6] and E...

Some Separability Problems in the Plane

Some combinatorial problems of importance

We discuss two combinatorial problems for which we have only sub-optimal solutions. We describe known solutions and we explain why better solutions would be of importance to Cryptography. The areas of application are in improving the ef-ciency of Zero-Knowledge proofs, in relating the quadratic residuosity problem to the problem of integer factorization, and in analyzing some cryptographic sche...

Combinatorial and computational problems about points in the plane

We study three problems in combinatorial geometry. The problems investigated are con ict-free colorings of point sets in the plane with few colors, polychromatic colorings of the vertices of rectangular partitions in the plane and in higher dimensions and polygonalizations of point sets with few re ex points. These problems are problems of discrete point sets, the proofs are of combinatorial av...

Some Problems in Combinatorial Number Theory

We state and discuss various problems in the general area of arithmetic combinatorics and recent developments related to the ‘sum-product phenomenon’ in the ring of integers, the real and complex numbers and finite fields. In particular, we discuss applications and connections to the theory of exponential sums, Burgess’ estimate, the subspace theorem and to Szemeredi-Trotter type results. In re...