Related papers: On intersection of simply connected sets in the pl…

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection…

Three lines are concurrent if they intersect at a single point. In this paper I prove that if $F$ is a bounded family of compact connected sets in the plane, such that every three sets in $F$ can be pierced by a single line, then there…

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of…

We study the sets of planes in an even dimensional real vector space $V$ which are simultaneously stabilised by a pair of complex structures on $V$. We completely describe these sets of planes for pairs of orthogonal complex structures.…

If two closed Jordan curves in the plane have precisely one point in common, then it is called a {\em touching point}. All other intersection points are called {\em crossing points}. The main result of this paper is a Crossing Lemma for…

Given S_1, a finite set of points in the plane, we define a sequence of point sets S_i as follows: With S_i already determined, let L_i be the set of all the line segments connecting pairs of points of the union of S_1,...,S_i, and let…

Given a finite set $ S $ of points, we consider the following reconfiguration graph. The vertices are the plane spanning paths of $ S $ and there is an edge between two vertices if the two corresponding paths differ by two edges (one…

We consider the following problem: Given a set $S$ of $n$ distinct points in the plane, how many edge-disjoint plane straight-line spanning paths can be drawn on $S$? Each spanning path must be crossing-free, but edges from different paths…

We prove that if two subsets ${A}$ and ${B}$ of the plane are connected, ${A}$ is bounded, and the Euclidean distance $\rho({A},{B})$ between ${A}$ and ${B}$ is greater than zero, then for every positive $\varepsilon<\rho({A},{B})$, the…

Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $\sigma$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in \sigma} A_i \neq \emptyset$. Let…

We prove that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines which have…

Two subsets $A, B$ of the plane are betweenness isomorphic if there is a bijection $f\colon A\to B$ such that, for every $x,y,z\in A$, the point $f(z)$ lies on the line segment connecting $f(x)$ and $f(y)$ if and only if $z$ lies on the…

Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a…

A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions…

A connected topological space is said to be widely-connected if each of its non-degenerate connected subsets is dense in the entire space. The object of this paper is the construction of widely-connected subsets of the plane. We give a…

Say that a subset S of the plane is a "circle-center set" if S is not a subset of a line, and whenever we choose three noncollinear points from S, the center of the unique circle through those three points is also an element of S. A problem…

We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R_1,...,R_n in the plane and finite point sets P_i subset of R_i for i=1,...,n, are there non-crossing connectors y_i for…

Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible…

Let $S$ be a finite set of points in the plane and let $\mathcal{T}(S)$ be the set of intersection points between pairs of lines passing through any two points in $S$. We characterize all configurations of points $S$ such that iteration of…

We show that a simply connected stable plane with connected lines is isomorphic to an open subplane of a classical projective plane (i.e., a plane over the real or complex numbers, the quaternions or the octonions) if it has that property…