- Full text PDF available (45)
- This year (4)
- Last 5 years (17)
- Last 10 years (31)
Journals and Conferences
We prove almost tight bounds on incidences between points and k-dimensional varieties of bounded degree in R. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.
A 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 ε(H) > 0 such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least n? We answer this question in the affirmative for a special class… (More)
We present an elementary proof that if A is a finite set of numbers, and the sumset A+G A is small, |A+G A| ≤ c|A|, along a dense graph G, then A contains k-term arithmetic progressions.
We prove that the sumset or the productset of any finite set of real numbers, A, is at least |A|4/3−ε, improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, E(A, A).
For any coloring of the N×N grid using less than log log n colors, one can always find a monochromatic equilateral right triangle, a triangle with vertex coordinates (x, y), (x + d, y), and (x, y + d). AMS Mathematics Subject Classification: 05D10
In this paper we give incidence bounds for arrangements of curves in Fq . As an application, we prove a new result that if (x, f(x)) is a Sidon set then either A+A or f(A)+ f(A) should be large. The main goal of the paper is to illustrate the use of graph spectral techniques in additive combinatorics. This is an extended version of the talks I gave in the… (More)
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.
It is shown that a homogeneous set of n points in the d-dimensional Euclidean space determines at least Ω(n2d/(d 2+1)/ log n) distinct distances for a constant c(d) > 0. In three-space, the above general bound is slightly improved and it is shown that a homogeneous set of n points determines at least Ω(n.6091) distinct distances.
Given a system S of simple continuous curves (``strings'') in the plane, we can define a graph GS as follows. Assign a vertex to each curve, and connect two vertices by an edge if and only if the corresponding two curves intersect. GS is called the intersection graph of S. Not every graph is an intersection graph of a system of curves [EET76] (see Fig. 1… (More)