József Solymosi

An Incidence Theorem in Higher Dimensions

We prove almost tight bounds on the number of incidences between points and k-dimensional varieties of bounded degree in Rd. Our main tools are the polynomial ham sandwich theorem and induction on both the dimension and the number of points.

Ramsey-type theorems with forbidden subgraphs

Dedicated to the memory of Paul ErdősA graph is called H-free if it contains no induced copy of H. We discuss the following question raised by Erdős and Hajnal. Is it true that for every graph H, there exists an such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least ? We answer this question in the affirmative for a special class of graphs, and give an equivalent reformulation for tournaments. In order to prove the equivalence, we establish several Ramsey type results for tournaments.

Distinct Distances in the Plane

It is shown that every set of n points in the plane has an element from which there are at least cn6/7 other elements at distinct distances, where c>0 is a constant. This improves earlier results of Erdős, Moser, Beck, Chung, Szemerédi, Trotter, and Székely.

On the Number of Sums and Products

A new lower bound on

Note on a Generalization of Roth’s Theorem

\max \{|A+A|,|A\cdot A|\}

A Note on a Question of Erdös and Graham

is given, where

Distinct distances on two lines

A

Near optimal bounds for the Erdos distinct distances problem in high dimensions

is a finite set of complex numbers.

Crossing Patterns of Segments

We give a simple proof that for sufficiently large N, every subset of of size[N 2]of size at least δN 2 contains three points of the form (a,b), (a + d, b), (a, b + d).

Polynomials vanishing on grids: The Elekes-Rónyai problem revisited

We give a quantitative proof that, for sufficiently large