In GIS (Geographic Information Systems), Cartography, Social Network Analysis, Robotics and elsewhere, we often face structural questions about collections of complex geometric objects. For example, given some closed curves in the plane and the regions enclosed by them (e.g. the coverage area of transmission towers), we need to study their mutual relations, interference, and other combinatorial properties.
Falconer's conjecture is a continuous analogue of Erdős' distinct distances problem in geometric measure theory. It states that a set of points that is large in its Hausdorff dimension must determine a set of distances that is large in measure. The radial projection from a point y maps x to the line containing x and y. We prove various bounds on the size of the projection sets, which can be used in certain ranges on the the variant of Erdős' problem over a finite field, as demonstrated by Orponen.
By improving upon previous estimates on a problem posed by Leo Moser, we proved the old conjecture of Erdős, according to which the density of any measurable planar set avoiding unit distances is strictly less than 1/4.
A celebrated result of Alon states that, under proper divisibility assumptions, any string (``necklace'') of kn letters (``beads'') of t different kinds can be cut at t(k-1) places such that the resulting pieces can be grouped into k groups, each of which contains the same number of letters of any given type. Suppose that the letters are ordered uniformly at random. We work out the typical number of cuts needed, which is often much lower than in the worst case bound.
It was one of the first successful applications of the Probabilistic Method about 65 years ago that Erdős constructed graphs of arbitrary high girth and chromatic number. Effective constructions with this property play important roles in combinatorics. We build such graphs, whose vertices are polygonal chains in the plane, consisting of at most 3 segments, two of them being connected by an edge if and only if they are disjoint.
We solved a topological variant of Heilbronn's problem, one of the oldest questions in combinatorial geometry, concerning the area of the smallest triangle determined by n points in a unit square. This has interesting ramifications in Graph Drawing and Representation.
We wrote a survey paper in Hungarian, intended for talented high school and college students, on the spectacular recent breakthroughs in Ramsey theory, concerning diagonal and off-diagonal Ramsey numbers.
We described, discussed, and generalized a famous mathematics competition problem concerning reconfiguration of point sets in the plane and in higher dimensions. We published an expository paper in the American Mathematical Monthly on how to approach these questions using algebraic techniques.
