The project deals with the study of central problems in combinatorics and discrete mathematics. It covers specially three research axes: geometric combinatorics, algebraic combinatorics and probabilistic combinatorics. More precisely, the main objectives are the following: - Freiman conjecture on doubling and volume, its consequences in additive combinatorics and its interplay with the so-called polynomial Freiman-Ruzsa conjecture. - Generalization of graph invariants (chromatic polynomial and symmetric chromatic polynomial), as well as their interactions with matroid theory. - New enumerative and probabilistic results for graph families with topological obstructions, covering bipartite graph classes and regular graphs, among others. - Study of problems in incidence geometry, more particularly in relation to the MDS conjecture and to the Silvester problem for dimension higher than 4. - Study of new logic laws for random discrete structures, including rooted graphs, perfect graphs and permutations. - Analysis of the distance chromatic number of cartesian products of graphs and extensions of the sandwich theorem for non-monotone parameters. The study of these problems involves the development of new geometric, algebraic and probabilistic techniques, as well as the interaction of these areas and the use of new ones, including analytic, topological an number theoretical tools. Building a solid ground for this set of techniques is one of the side objectives of the proposal.
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia