Related papers: A user's guide to the topological Tverberg conject…
We discuss here analogs of van der Waerden and Tverberg permanent conjectures for haffnians on the convex set of matrices whose extreme points are symmetric permutation matrices with zero diagonal.
In this paper we present a combinatorial proof of Selberg's integral formula. We start by giving a bijective proof of a Theorem about the number of topological orders of a certain related directed graph. Selberg's Integral Formula then…
The coloured Tverberg theorem was conjectured by B\'ar\'any, Lov\'{a}sz and F\"uredi and asks whether for any d+1 sets (considered as colour classes) of k points each in R^d there is a partition of them into k colourful sets whose convex…
The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are…
We compute a primary cohomological obstruction to the existence of an equipartition for j mass distributions in R^d by two hyperplanes in the case 2d-3j = 1. The central new result is that such an equipartition always exists if d=6 2^k +2…
We give a new simple geometric proof that any seven points in the plane have four Tverberg partitions into three sets. This is the only confirmed non-trivial case of Sierksma's conjecture. Earlier proofs, by Stephan Hell, relied on…
An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. Seymour [On multicolourings of cubic graphs, and conjectures of Fulkerson and Tutte.~\emph{Proc.~London…
We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than…
An $r$-graph is an $r$-regular graph where every odd set of vertices is connected by at least $r$ edges to the rest of the graph. Seymour conjectured that any $r$-graph is $r+1$-edge-colorable, and also that any $r$-graph contains $2r$…
We survey two decades of work on the (sequential) topological complexity of configuration spaces of graphs (ordered and unordered), aiming to give an account that is unifying, elementary, and self-contained. We discuss the traditional…
Let $r \ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Gy\H{o}ri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Tur\'{a}n graph on $n$ vertices and $r - 1$ parts,…
This survey paper deals with upper and lower bounds on the number of $k$-matchings in regular graphs on $N$ vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower…
We prove that if a pure simplicial complex of dimension d with n facets has the least possible number of (d-1)-dimensional faces among all complexes with n faces of dimension d, then it is vertex decomposable. This answers a question of J.…
Let $p$ be an odd prime and $r\geq 1$. Suppose that $\alpha$ is a $p$-adic integer with $\alpha\equiv2a\pmod p$ for some $1\leq a<(p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that…
We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order…
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…
Tverberg's theorem asserts that every (k-1)(d+1)+1 points in R^d can be partitioned into k parts, so that the convex hulls of the parts have a common intersection. Calder and Eckhoff asked whether there is a purely combinatorial deduction…
Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean d-space that guarantees any such point set admits a…
In the first part of this paper, we give a new analytical proof of a theorem of C. Sabbah on integrable deformations of meromorphic connections on $\mathbb P^1$ with coalescing irregular singularities of Poincar\'e rank 1, and generalizing…
Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the…