Related papers: Balanced lines in two-coloured point sets

Let $S$ be a set of $r$ red points and $b=r+2d$ blue points in general position in the plane, with $d\geq 0$. A line $\ell$ determined by them is said to be balanced if in each open half-plane bounded by $\ell$ the difference between the…

Given a set of red and blue points in the plane, a bichromatic line is a line containing at least one red and one blue point. We prove the following conjecture of Kleitman and Pinchasi (unpublished, 2003). Let P be a set of n red, and n or…

Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several…

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $\alpha \in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly…

We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way…

Let $S=R\cup B$ be a point set in the plane in general position such that each of its elements is colored either red or blue, where $R$ and $B$ denote the points colored red and the points colored blue, respectively. A quadrilateral with…

Let $P$ be a set of $n$ green and $n - k$ red points in $\mathbb{C}^2$. A line determined by $i$ green and $j$ red points such that $i + j \ge 2$ and $|i - j| \le r$ is called \emph{r-equichromatic}. We establish lower bounds for…

Suppose that $nk$ points in general position in the plane are colored red and blue, with at least $n$ points of each color. We show that then there exist $n$ pairwise disjoint convex sets, each of them containing $k$ of the points, and each…

Given a point set $S$ in a projective plane $\Pi_q$ of order $q$, each line $\ell$ determines a secant size $|S\cap \ell|$. We study how balanced the secant-size distribution can be for the line set $\mathcal{L}$ of the plane, in other…

An old question in Euclidean Ramsey theory asks, if the points in the plane are red-blue coloured, does there always exist a red pair of points at unit distance or five blue points in line separated by unit distances? An elementary proof…

Given a set R of red points and a set B of blue points in the plane, the Red-Blue point separation problem asks if there are at most k lines that separate R from B, that is, each cell induced by the lines of the solution is either empty or…

An important theorem of Beck says that any point set in the Euclidean plane is either ``nearly general position'' or ``nearly collinear'': there is a constant C>0 such that, given n points in the plane with at most r$ of them collinear, the…

By a polygonization of a finite point set $S$ in the plane we understand a simple polygon having $S$ as the set of its vertices. Let $B$ and $R$ be sets of blue and red points, respectively, in the plane such that $B\cup R$ is in general…

We prove that for every integer $k$, every finite set of points in the plane can be $k$-colored so that every half-plane that contains at least $2k-1$ points, also contains at least one point from every color class. We also show that the…

We consider the following problem: Let $\mathcal{L}$ be an arrangement of $n$ lines in $\mathbb{R}^3$ colored red, green, and blue. Does there exist a vertical plane $P$ such that a line on $P$ simultaneously bisects all three classes of…

Given n red and n blue points in general position in the plane, it is well-known that there is a perfect matching formed by non-crossing line segments. We characterize the bichromatic point sets which admit exactly one non-crossing…

We show that each set of $n\ge 2$ points in the plane in general position has a straight-line matching with at least $(5n+1)/27$ edges whose segments form a connected set, and such a matching can be computed in $O(n \log n)$ time. As an…

Consider a bicolored point set $P$ in general position in the plane consisting of $n$ blue and $n$ red points. We show that if a subset of the red points forms the vertices of a convex polygon separating the blue points, lying inside the…

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of…

We prove that if a finite point set in real space does not have too many points on a plane, then it spans a quadratic number of ordinary lines. This answers the real case of a question of Basit, Dvir, Saraf, and Wolf. It shows that there is…