Related papers: Stabbing Pairwise Intersecting Disks by Five Point…

These notes are intended as an easy-to-read supplement to part of the background material presented in my talks on enumerative geometry. In particular, the numbers $n_3$ and $n_4$ of plane rational cubics through eight points and of plane…

A disk graph is an intersection graph of disks in the Euclidean plane, where the disks correspond to the vertices of the graph and a pair of vertices are adjacent if and only if their corresponding disks intersect. The problem of…

Given a set of unit-disks in the plane with union area $A$, what fraction of $A$ can be covered by selecting a pairwise disjoint subset of the disks? Rado conjectured 1/4 and proved $1/4.41$. Motivated by the problem of channel-assignment…

When the plane is pie-sliced in $n\leq 4$ parts (with nonempty interior and common vertex at the origin) our main result provides a sufficient condition for any map $L$, that is continuous and piecewise linear relatively to this slicing, to…

Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P. This confirms, for large n, a conjecture of…

We consider the minimum number of lines $h_n$ and $p_n$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ chessboard. Determining these values can also be interpreted as a strengthening of the classical plank…

We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…

Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoi\v{c}i\'{c} (2003) showed that the limiting set is dense…

Two arrangements with the same combinatorial intersection lattice but whose complements have different fundamental groups are called a Zariski pair. This work finds that there are at most nine such pairs amongst all ten line arrangements…

Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k…

We present a novel feasibility criteria for the finite intersection of convex sets given by inequalities. This criteria allows us to easily assert the feasibility by analyzing the unconstrained minimum of a speci?fic convex function, that…

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.…

Alternative novel measures of the distance between any two partitions of a n-set are proposed and compared, together with a main existing one, namely 'partition-distance' D(.,.). The comparison achieves by checking their restriction to…

Discs form a compact packing of the plane if they are interior disjoint and the graph which connects the center of mutually tangent discs is triangulated. There is only one compact packing by discs all of the same size, called hexagonal…

We consider hard-disc mixtures with disc sizes within ratio $\sqrt{2}-1$, that is, the small disc exactly fits in the hole between four large discs. For each prescribed stoichiometry of large and small discs, the densest packings are…

Two dimensional particle disks in proximity to a planet are numerically integrated to determine when a planet in a circular orbit can truncate a particle disk. Collisions are treated by giving each particle a series of velocity…

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of…

A point set $P \subset {\Bbb{R}}^d$ is {\it separated} if the minimum distance between any two points in $P$ is at least $1$. For $d \ne 4,5,$ we determine, for every $t_1,t_2 \ge 1$, and for $n$ at least a suitable $n_d$, the maximum…

We prove that Multicut in directed graphs, parameterized by the size of the cutset, is W[1]-hard and hence unlikely to be fixed-parameter tractable even if restricted to instances with only four terminal pairs. This negative result almost…

Dirac and Motzkin conjectured that any set X of $n$ non-collinear points in the plane has an element incident with at least $\lceil \frac{n}{2} \rceil$ lines spanned by X. In this paper we prove that any set X of $n$ non-collinear points in…