Related papers: Unsplittable coverings in the plane

For each m>=1 and k>=2, we construct a graph G=(V,E) with \omega(G)=m such that max_{1\leq i\leq k} \omega(G[V_i])=m for arbitrary partition V=V_1\cup...\cup V_k, where \omega(G) is the clique number of G and G[V_i] is the induced subgraph…

Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset…

For every convex disk $K$ (a convex compact subset of the plane, with non-void interior), the packing density $\delta(K)$ and covering density $\vartheta(K)$ form an ordered pair of real numbers, {\em i.e.}, a point in ${\mathbb R}^2$. The…

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every…

Let $S$ be a set of $n$ points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of $S$ is less than $Kn^2$ for some $K=o(n^{\frac{1}{7}})$…

A clutter is \emph{clean} if it has no delta or the blocker of an extended odd hole minor, and it is \emph{tangled} if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in…

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be…

The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics. It has only been exactly solved for the special case of dimer coverings in two dimensions. In earlier work, Stanley proved a…

$\delta$-Covering, for some covering range $\delta>0$, is a continuous facility location problem on undirected graphs where all edges have unit length. The facilities may be positioned on the vertices as well as on the interior of the…

A similarity structure on a connected manifold M is a Riemannian metric on its universal cover such that the fundamental group of M acts by similarities. If the manifold M is compact, we show that the universal cover admits a de Rham…

Let $M$ be a 3-manifold. Every knotted (embedded) surface in $M \times \R$ can be moved via an ambient isotopy in such a way that its projection into $M$ is a generic surface. A surface is generic if every point on it is either a regular,…

We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds…

In this paper, we study necessary and sufficient conditions for the existence of categorical universal coverings using open covers of a given space $X$. As some applications, first we present a generalized version of the Shelah Theorem…

Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\geq 3$. We will prove that for all $m \in \mathbb{N}$, there exists $c^{*}(m)>0$, which depends on $g$, such that if $0<c<c^{*}(m)$, $(M,g)$ contains at least $m$ many closed $c$-CMC…

It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be…

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular…

We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by…

In this paper, we use the Mordukhovich derivatives to precisely find the covering constants for the metric projection operator onto nonempty closed and convex subsets in uniformly convex and uniformly smooth Banach spaces. We consider three…

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an…

This work revolves around the two following questions: Given a convex body $C\subset\mathbb{R}^d$, a positive integer $k$ and a finite set $S\subset\mathbb{R}^d$ (or a finite Borel measure $\mu$ on $\mathbb{R}^d$), how many homothets of $C$…