Periodic Reporting for period 2 - GeoScape (From Geometry to Combinatorics and Back: Escaping the Curse of Dimensionality)
Période du rapport: 2022-03-01 au 2023-08-31
Asymptotic results in extremal and probabilistic combinatorics have proved to be powerful tools in the structural and algorithmic analysis of huge networks such as the internet graph, brain maps, social networks, and integrated circuits.
Our main objective was to study hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications.
In the present project, we have aimed to explore the following direction: to develop and apply geometric techniques to settle important special cases of notoriously difficult combinatorial problems on
(1) bounded degree semi-algebraic graphs and hypergraphs,
(2) graphs and hypergraphs of bounded VC-dimension,
(3) ordered graphs, 0-1 matrices, and graphs embedded in the plane or in other surfaces.
One can model some brain maps as intersection graphs of geometric objects, as described in the project proposal. Disjointness graphs are defined very similarly. We proved that disjointness graphs of segments in the space are chi-bounded, that is, the chromatic number is bounded in terms of the clique number. However, we show that it is not the case for polygonal chains of length two. For polygonal chains of length three we have an even stronger result, the girth and the chromatic number can be arbitrarily large simultaneously.
Ramsey theory is one of the most actively studied parts of Combinatorics. The first questions on Ramsey numbers, in a slightly different form, were proposed by Schur. Erdős offered a prize for deciding whether Ramsey numbers grow at most exponentialy with the number of colors. We proved that the answer was yes for graphs with bounded dual Vapnik-Chervonenkis dimension.
The study of drawings is motivated by several applications, including CAD, cartography and circuit schematics. We extended the Crossing Lemma in two directions. If no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in G is the same order as in the Crossing Lemma. If nonparallel edges are allowed to cross any number of times, then the number of crossings in G has a different order of magnitude. We managed to improve the coefficient for some other versions, in particular, for the odd-crossing number, where we only count pairs of edges that cross an odd number of times.
We successfully applied the ideas sketched in the proposal to prove a slightly weaker version of the celebtated Sunflower Conjecture of Erdős and Rado. We came close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension. We also verified the Erdős-Rado conjecture for families of bounded Littlestone dimension and for some geometrically defined set systems.
Our main objective was to study hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications.
In the present project, we have aimed to explore the following direction: to develop and apply geometric techniques to settle important special cases of notoriously difficult combinatorial problems on
(1) bounded degree semi-algebraic graphs and hypergraphs,
(2) graphs and hypergraphs of bounded VC-dimension,
(3) ordered graphs, 0-1 matrices, and graphs embedded in the plane or in other surfaces.
One can model some brain maps as intersection graphs of geometric objects, as described in the project proposal. Disjointness graphs are defined very similarly. We proved that disjointness graphs of segments in the space are chi-bounded, that is, the chromatic number is bounded in terms of the clique number. However, we show that it is not the case for polygonal chains of length two. For polygonal chains of length three we have an even stronger result, the girth and the chromatic number can be arbitrarily large simultaneously.
Ramsey theory is one of the most actively studied parts of Combinatorics. The first questions on Ramsey numbers, in a slightly different form, were proposed by Schur. Erdős offered a prize for deciding whether Ramsey numbers grow at most exponentialy with the number of colors. We proved that the answer was yes for graphs with bounded dual Vapnik-Chervonenkis dimension.
The study of drawings is motivated by several applications, including CAD, cartography and circuit schematics. We extended the Crossing Lemma in two directions. If no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in G is the same order as in the Crossing Lemma. If nonparallel edges are allowed to cross any number of times, then the number of crossings in G has a different order of magnitude. We managed to improve the coefficient for some other versions, in particular, for the odd-crossing number, where we only count pairs of edges that cross an odd number of times.
We successfully applied the ideas sketched in the proposal to prove a slightly weaker version of the celebtated Sunflower Conjecture of Erdős and Rado. We came close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension. We also verified the Erdős-Rado conjecture for families of bounded Littlestone dimension and for some geometrically defined set systems.
The disjointness graph G of a set of segments S is a graph whose vertex set is S and two vertices are adjacent if and only if the corresponding segments are disjoint.
We proved that the chromatic number of such a graph G is polynomially bounded by the clique number of G.
We showed that computing the clique number and the chromatic number for disjointness graphs of lines in space are NP-hard tasks.
We constructed families of arcs, whose disjointness graphs are triangle-free, but whose chromatic numbers are arbitrarily large.
Let G be a multigraph drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior.
Pach and Tóth extended the Crossing Lemma of Ajtai et al. and Leighton by showing that if no two adjacent edges cross and every pair of nonadjacent edges
cross at most once, then the number of edge crossings in G is of the same order as in the Crossing Lemma.
The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times.
We proved that in this case the number of crossings in G has a different order of magnitude.
A “cornerstone” of the proposed research was to approach notoriously difficult unsolved problems for bounded-dimensional set systems.
We successfully used this idea to prove a slightly weaker version the famous Sunflower Conjecture of Erdős and Rado.
We came close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension.
We also verified the Erdős-Rado conjecture for families of bounded Littlestone dimension and for some geometrically defined set systems.
We proved that the chromatic number of such a graph G is polynomially bounded by the clique number of G.
We showed that computing the clique number and the chromatic number for disjointness graphs of lines in space are NP-hard tasks.
We constructed families of arcs, whose disjointness graphs are triangle-free, but whose chromatic numbers are arbitrarily large.
Let G be a multigraph drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior.
Pach and Tóth extended the Crossing Lemma of Ajtai et al. and Leighton by showing that if no two adjacent edges cross and every pair of nonadjacent edges
cross at most once, then the number of edge crossings in G is of the same order as in the Crossing Lemma.
The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times.
We proved that in this case the number of crossings in G has a different order of magnitude.
A “cornerstone” of the proposed research was to approach notoriously difficult unsolved problems for bounded-dimensional set systems.
We successfully used this idea to prove a slightly weaker version the famous Sunflower Conjecture of Erdős and Rado.
We came close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension.
We also verified the Erdős-Rado conjecture for families of bounded Littlestone dimension and for some geometrically defined set systems.
We expect to prove theorems related to geometric variants of the following classical graph-theoretic problem of Erdős, Gallai, and Rogers. Given a K_r-free graph on n vertices and an integer s < r, at least how many vertices can we find such that the subgraph induced by them is K_s-free?
In particular, we will concentrate on string graphs.
Schmidt and Tuller posed a conjecture concerning optimal packings and coverings of integers by translates of a given three-element set.
We expect to confirm their conjecture and relate it to several other problems in combinatorics.
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 want to prove 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 will present a new recursive construction to combine Ramsey sets, which is expected to lead to improved lower bounds on Euclidean Ramsey numbers. We expect to obtain similar results for Ramsey numbers of higher-dimensional regular simplices, as well as for similar problems in other norms.
We plan to prove that any normed space can be two-colored such 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 expect to prove a sharp result for the number of maximal strongly independent sets in a 3-uniform hypergraph, analogous to the Moon-Moser theorem.
We expect to prove that the Helly numbers of exponential lattices are finite, and determine their exact values in some instances. In particular, we plan to solve a problem posed by Dillon (2021).
We plan to give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We might show that every graph can be
realized as a total coalition graph.
We try to answer 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 attack the colorful vector balancing problem and expect to significantly strengthen the estimate of Bárány and Grinberg for general norms. We combine linear algebra and probabilistic methods with a Gaussian random walk
argument.
In particular, we will concentrate on string graphs.
Schmidt and Tuller posed a conjecture concerning optimal packings and coverings of integers by translates of a given three-element set.
We expect to confirm their conjecture and relate it to several other problems in combinatorics.
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 want to prove 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 will present a new recursive construction to combine Ramsey sets, which is expected to lead to improved lower bounds on Euclidean Ramsey numbers. We expect to obtain similar results for Ramsey numbers of higher-dimensional regular simplices, as well as for similar problems in other norms.
We plan to prove that any normed space can be two-colored such 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 expect to prove a sharp result for the number of maximal strongly independent sets in a 3-uniform hypergraph, analogous to the Moon-Moser theorem.
We expect to prove that the Helly numbers of exponential lattices are finite, and determine their exact values in some instances. In particular, we plan to solve a problem posed by Dillon (2021).
We plan to give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We might show that every graph can be
realized as a total coalition graph.
We try to answer 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 attack the colorful vector balancing problem and expect to significantly strengthen the estimate of Bárány and Grinberg for general norms. We combine linear algebra and probabilistic methods with a Gaussian random walk
argument.