Related papers: Tur\'an and Ramsey problems for alternating multil…
A new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties…
The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine…
We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we…
We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemer\'edi's theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we…
The article primarily surveys work that followed from the formulas discovered by Avramov and Iyengar in 2008, which permit one to compute certain Hochschild homology and cohomology modules as expressions involving dualizing complexes. One…
There are numerous combinatorial objects associated to a Grassmannian permutation $w_\lambda$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations…
In this paper, we provide an alternative appraoch to an expectaion of F\"assler et al [J. Geom. Anal. 2016] by showing that a metrically quasiregular mapping between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$…
Classical Ramsey theory has successfully extended to relational structures, yielding a wealth of results that have profoundly influenced other areas of mathematics. Interestingly, the same development has not occurred in the case of dual…
Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line…
We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and…
We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…
Metric Ramsey theory is concerned with finding large well-structured subsets of more complex metric spaces. For finite metric spaces this problem was first studies by Bourgain, Figiel and Milman \cite{bfm}, and studied further in depth by…
In this paper, we study a problem posed by Furstenberg on intersections between $\times 2, \times 3$ invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used…
The $r$-expansion $G^+$ of a graph $G$ is the $r$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a vertex subset of size $r-2$ disjoint from $V(G)$ such that distinct edges are enlarged by disjoint subsets. Let…
This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by…
In contrast to the abundance of "direct" Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a…
In this paper we investigate iteration of maps on lattices and the corresponding polynomial-like iterative equation. Since a lattice need not have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point…
In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs $\mathcal{H}$ in $K_n$, defined as the minimum number of colours needed to colour the edges of $K_n$ so that every copy of a graph $H\in…
Ramsey theory is the study of conditions under which mathematical objects show order when partitioned. Ramsey theory on the integers concerns itself with partitions of $[1,n]$ into $r$ subsets and asks the question whether one (or more) of…
We consider two-dimensional N=(2,2) supersymmetric gauge theory on discretized Riemann surfaces. We find that the discretized theory can be efficiently described by using graph theory, where the bosonic and fermionic fields are regarded as…