Related papers: Disjoint axis-parallel segments without a circumsc…
In this note we show that unbounded convex polygons with nonparallel unbounded edges are polynomial images of ${\mathbb R}^2$, whereas their interiors are polynomial images of ${\mathbb R}^3$
For every integer $\ell$, we construct a cubic 3-vertex-connected planar bipartite graph $G$ with $O(\ell^3)$ vertices such that there is no planar straight-line drawing of $G$ whose vertices all lie on $\ell$ lines. This strengthens…
The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |,…
We construct, in even spacetime dimensions, a family of singularity-free Kerr-Anti-de Sitter-like black holes with negatively curved cross-sections of conformal infinity and non-spherical cross-sections of horizons.
There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already…
In this paper, six constructions of difference families are presented. These constructions make use of difference sets, almost difference sets and disjoint difference families, and give new point of views of relationships among these…
A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique…
A famous configuration of 27 lines on a non-singular cubic surface in $\mathbb P^3$ contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We study such six-line configurations in the case…
Three disjoint rays in euclidean 3-space form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise…
introduce {\sc Planar Disjoint Paths Completion}, a completion counterpart of the Disjoint Paths problem, and study its parameterized complexity. The problem can be stated as follows: given a, not necessarily connected, plane graph $G,$ $k$…
In this paper, we derive a formula to express the maximum number of non-intersecting diagonals of arbitrary length that can be drawn in n x n square arrays, where n is a multiple of l+1.
A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show…
We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does…
We give a non-Paschian plane based on the property of betweenness which cannot be derived from an ordering of the points of a line. In this model there is no possibility to define the congruence of segments but we can define angle, triangle…
By examining the 3 surface angles which exist at any of the 8 vertices of a Diophantine parallelepiped, and classifying them by the appearance of a right angle, it is discovered that 5 unique classes of Diophantine parallelepipeds exist. It…
In this work, we carry out structural and algorithmic studies of a problem of barrier forming: selecting theminimum number of straight line segments (barriers) that separate several sets of mutually disjoint objects in the plane. The…
Descending plane partitions, alternating sign matrices, and totally symmetric self-complementary plane partitions are equinumerous combinatorial sets for which no explicit bijection is known. In this paper, we isolate a subset of descending…
We prove that for all positive integers $n$ and $k$, there exists an integer $N = N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and $\mathcal{J}_U$ is the set of all line segments in direction $u$…
We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the $1/k$ grids. The intersection of two cells can be…
We construct a non-full exceptional collection of maximal length consisting of line bundles on the blow-up of the projective plane in 10 points in general position. This provides a counterexample to a conjecture of Kuznetsov and to a…