Periodic Reporting for period 3 - GeoScape (From Geometry to Combinatorics and Back: Escaping the Curse of Dimensionality)
Periodo di rendicontazione: 2023-09-01 al 2025-02-28
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.
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.
We proved a 50-year old conjecture of Grünbaum, according to which any arrangement of n pairwise intersecting circles in the plane determine at most 2n-2 digonal cells.
We showed that every string graph of large chromatic number contains a large complete subgraph. This can be found by a polynomial time algorithm.
A linear ordering of the vertex set of a hypergraph H is called an agreeing linear order, provided that the vertices of every hyperedge lie between its two boundary vertices. We proved the following Helly-type theorem: if there is an agreeing linear order on the vertex set of every subhypergraph of H with at most 2r−2 vertices, then there is an agreeing linear order on the vertex set of H.
We presented a new recursive construction to combine Ramsey sets, which is expected to lead to improved lower bounds on Euclidean Ramsey numbers. We proved that any normed space can be two-colored with the property that all sufficiently long unit arithmetic progressions contain points of both colors.
A set S of vertices in a hypergraph is strongly independent if every hyperedge shares at most one vertex with S. We proved a sharp result for the number of maximal strongly independent sets in a 3-uniform hypergraph, analogous to the Moon-Moser theorem for graphs.
We proved that the Helly numbers of exponential lattices are finite, and in some instances we also determined their exact values. In particular, we solved a problem posed by Dillon (2021).
We answered a question of Letzter and Snyder, by proving that for large enough k, any n-vertex graph G with minimum degree at least n/(2k−1) and without odd cycles of length less than 2k + 1 is 3-colourable.
We attacked the colorful vector balancing problem and significantly strengthened the estimate of Bárány and Grinberg for general norms. We combined linear algebra and probabilistic methods with a Gaussian random walk argument.
We solved the finite field analog of a conjecture due to Liu and Orponen on the exceptional set of radial projections of a set of dimensions between d-2 and d-1.
We quantified the number of disjoint faces in complete simple topological graphs. In particular, we proved that a complete simple topological graph drawn in the plane must generate a linear number of quadrilaterals, which is best possible.
We constructed family of subsets of an n-element set without odd sunflowers of size 1.5^n. We also characterized minimal odd-sunflowers of triples.
Greatly improving earlier results, we constructed a red/blue coloring of the n-dimensional Euclidean plane without 3 red points separated by unit distances and without 1177 blue points on a line so that there is unit distance between the consecutive points.
We showed that every string graph of large chromatic number contains a large complete subgraph. This can be found by a polynomial time algorithm.
A linear ordering of the vertex set of a hypergraph H is called an agreeing linear order, provided that the vertices of every hyperedge lie between its two boundary vertices. We proved the following Helly-type theorem: if there is an agreeing linear order on the vertex set of every subhypergraph of H with at most 2r−2 vertices, then there is an agreeing linear order on the vertex set of H.
We presented a new recursive construction to combine Ramsey sets, which is expected to lead to improved lower bounds on Euclidean Ramsey numbers. We proved that any normed space can be two-colored with the property that all sufficiently long unit arithmetic progressions contain points of both colors.
A set S of vertices in a hypergraph is strongly independent if every hyperedge shares at most one vertex with S. We proved a sharp result for the number of maximal strongly independent sets in a 3-uniform hypergraph, analogous to the Moon-Moser theorem for graphs.
We proved that the Helly numbers of exponential lattices are finite, and in some instances we also determined their exact values. In particular, we solved a problem posed by Dillon (2021).
We answered a question of Letzter and Snyder, by proving that for large enough k, any n-vertex graph G with minimum degree at least n/(2k−1) and without odd cycles of length less than 2k + 1 is 3-colourable.
We attacked the colorful vector balancing problem and significantly strengthened the estimate of Bárány and Grinberg for general norms. We combined linear algebra and probabilistic methods with a Gaussian random walk argument.
We solved the finite field analog of a conjecture due to Liu and Orponen on the exceptional set of radial projections of a set of dimensions between d-2 and d-1.
We quantified the number of disjoint faces in complete simple topological graphs. In particular, we proved that a complete simple topological graph drawn in the plane must generate a linear number of quadrilaterals, which is best possible.
We constructed family of subsets of an n-element set without odd sunflowers of size 1.5^n. We also characterized minimal odd-sunflowers of triples.
Greatly improving earlier results, we constructed a red/blue coloring of the n-dimensional Euclidean plane without 3 red points separated by unit distances and without 1177 blue points on a line so that there is unit distance between the consecutive points.
We plan to make progress related to Albertson's conjecture, closely related to Hadwiger's conjecture, one of the central unsolved problems in structural graph theory. According to Albertson's conjecture, the crossing number of every graph whose chromatic number is k, is at least as large as the crossing number of a complete graph with k vertices. To achieve this, in some special cases we want to explore the Lescure-Meyniel conjecture.
We plan to investigate the possible number of edges of saturated partial plane embeddings of maximal n-vertex planar graphs. If sat_P(G) denotes the minimum number of edges of such a graph, then we want to show sat_P (G) < (3 − ϵ)n, for a positive ϵ.
We will study a generalization of the Ryser-Brualdi-Stein Conjecture to hypergraphs introduced by Aharoni, Charbit and Howard. The problem is to determine the largest rainbow matching, given a certain collection of matchings in a uniform hypergraph. We try to improve the best known lower and upper bounds for this question.
We study how many comparability subgraphs are needed to partition or cover the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs.
For interval graphs, an arbitrarily large number of comparability subgraphs might be necessary.
If no two segments of a drawing of graph G cross each other, then G is a plane graph. Over 20 years ago, Bose et al asked whether every complete geometric graph on n points can be partitioned into at most cn plane subgraphs, c<1. We expect to answer this question in the affirmative in the special case where the underlying point set is dense.
We plan to tackle the double cap conjecture that asks for the maximal density of a subset of the 2-sphere that avoids orthogonal vectors. We already have preliminary estimates that improve on all previous bounds.
We plan to investigate the possible number of edges of saturated partial plane embeddings of maximal n-vertex planar graphs. If sat_P(G) denotes the minimum number of edges of such a graph, then we want to show sat_P (G) < (3 − ϵ)n, for a positive ϵ.
We will study a generalization of the Ryser-Brualdi-Stein Conjecture to hypergraphs introduced by Aharoni, Charbit and Howard. The problem is to determine the largest rainbow matching, given a certain collection of matchings in a uniform hypergraph. We try to improve the best known lower and upper bounds for this question.
We study how many comparability subgraphs are needed to partition or cover the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs.
For interval graphs, an arbitrarily large number of comparability subgraphs might be necessary.
If no two segments of a drawing of graph G cross each other, then G is a plane graph. Over 20 years ago, Bose et al asked whether every complete geometric graph on n points can be partitioned into at most cn plane subgraphs, c<1. We expect to answer this question in the affirmative in the special case where the underlying point set is dense.
We plan to tackle the double cap conjecture that asks for the maximal density of a subset of the 2-sphere that avoids orthogonal vectors. We already have preliminary estimates that improve on all previous bounds.