Related papers: A solution to the Pyjama Problem

In 1985 Hopcroft, Joseph and Whitesides showed it is NP-complete to decide whether a carpenter's ruler with segments of given positive lengths can be folded into a line of at most a given length, such that the folded hinges alternate…

We study graphs coming from quadratic spaces over finite fields via orthogonality which generalize a recent result given by Bishnoi, Ihringer, and Pepe (2019). More precisely, we study the graph $\Gamma^{\square}(n,k,q)$ as follows: the…

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of…

The square peg problem asks whether every continuous curve in the plane that starts and ends at the same point without self-intersecting contains four distinct corners of some square. Toeplitz conjectured in 1911 that this is indeed the…

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel's circle problem (1959). The proof relies on a…

In 1989, Rota conjectured that, given any $n$ bases $B_1,\dots,B_n$ of a vector space of dimension $n$, or more generally a matroid of rank $n$, it is possible to rearrange these into $n$ disjoint transversal bases. Here, a transversal…

For $\Omega_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u E^{\gamma}_{\Omega_\e}(u)\] where \[ E^{\gamma}_{\Omega_\e}(u):=…

A star of n (n greater than or equal to 2) line segments (needles) of equal length with common endpoint and constant angular spacing is randomly placed onto a lattice which is the union of two families of equidistant lines in the plane with…

In this paper we deal with a classical problem in elementary number theory, namely repeating decimals. We show how the digits of the period of the decimal representation of any fraction $\frac{k}{m}$, where $k$ and $m$ are positive integers…

We define the parametric closure problem, in which the input is a partially ordered set whose elements have linearly varying weights and the goal is to compute the sequence of minimum-weight lower sets of the partial order as the weights…

We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $\Sigma^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact…

Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position. We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$ edges. Two vertices in…

Given a finite grid in $\mathbb{R}^2$, how many lines are needed to cover all but one point at least $k$ times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We…

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…

We examine non-linear sigma (plus Skyrme term) model in flat space, in particular with a twist, which comprises a twisted baby Skyrmion vortex solution with an added dependence on a twist term $mkz$, where $z$ is the vertical coordinate. We…

The Topological Tverberg Theorem claims that any continuous map of a (q-1)(d+1)-simplex to \R^d identifies points from q disjoint faces. (This has been proved for affine maps, for d=1, and if q is a prime power, but not yet in general.) The…

The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still open whether the Hirsch…

For any delta > 1 we construct a periodic and locally finite packing of the plane with ellipses whose delta-enlargement covers the whole plane. This answers a question of Imre B\'ar\'any. On the other hand, we show that if C is a packing in…

We conjecture that every oriented graph $G$ on $n$ vertices with $\delta ^+ (G) , \delta ^- (G) \geq 5n/12$ contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…