Related papers: A Crossing Lemma for Jordan Curves

By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a closed interval. We call a curve gamma in R^d at most k crossing if it intersects every hyperplane at most k times (counted with multiplicity). The at most d crossing…

Meyerson's Theorem says that all but at most 2 points of any Jordan loop are vertices of inscribed equilateral triangles. We show that for any Jordan loop there are uncountable many other triangle shapes for which this same result is true.…

A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each…

The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that…

A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at…

We generalize the joints problem to sets of varieties and prove almost sharp bound on the number of joints. As a special case, given a set of $N$ $2$-planes in $\mathbb{R}^6$, the number of points at which three $2$-planes intersect and…

This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is…

In this paper we prove that if $\gamma$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $\gamma$ in $\mathcal{C}^0$ as $t\to 0^+ $. Another perspective is that the…

Theorem converse to Jordan's curve theorem says that {\it if a compact set $K$ has two complementary domains in $R^{2}$, from each of which it is at every point accessible, it is a simple closed curve}. We show that the requirement of this…

The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant…

We show that if a graph $G$ with $n \geq 3$ vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then $G$ has at most $6n-12$ edges. This settles a conjecture of Pach, Radoi\v{c}i\'{c},…

Given $n$ points in the plane, a \emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice…

We prove that for every $\epsilon>0$ there exists $\delta>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-\epsilon)\frac{n^{2}}{2}$ pairs of curves…

A plane drawing of a graph is {\em cylindrical} if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The {\em cylindrical crossing…

Let $\mathcal{Q}_1$ and $\mathcal{Q}_2$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^n(\mathbb{F}_q)$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result…

Let $G$ be a drawing of a graph with $n$ vertices and $e>4n$ edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chv\'atal, Newborn,…

We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments.…

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from…

A curve \gamma in the plane is t-monotone if its interior has at most t-1 vertical tangent points. A family of t-monotone curves F is \emph{simple} if any two members intersect at most once. It is shown that if F is a simple family of n…

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with $n$ self-crossings requires…