Add to ORKGWe prove that for any point set <em>P</em> in the plane, a triangle <em>T</em>, and a positive integer <em>k</em>, there exists a coloring of <em>P</em> with <em>k</em> colors such that any homothetic copy of <em>T</em> containing at least 144<em>k</em><sup>8</sup> points of <em>P</em> contains at least one of each color. This is the first polynomial bound for range spaces induced by homothetic polygons.The only previously known bound for this problem applies to the more general case of octants in ℝ<sup>3</sup>, but is doubly exponential
AbstractGiven a set T of nonnegative integers, a T-coloring of a graph G is a labeling of the vertic...
A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignme...
To a set of n points in the plane, one can associate a graph that has less than n 2 vertices and has...
We prove that for any point set P in the plane, a triangle T, and a positive integer k, there exists...
Abstract. We prove that for any finite point set P in the plane, a triangle T, and a positive intege...
International audienceWe give new positive results on the long-standing open problem of geometric co...
We prove that every finite set of homothetic copies of a given convex body in the plane can be color...
Our point of departure is the following simple common generalisation of the Sylvester–Gallai theorem...
AbstractThe 512 points of an 8 × 8 × 8 cubical array of lattice points in 3-space can be colored in ...
Let <em>P</em> be a set of <em>n</em> points in general position in the plane. We study the chromati...
AbstractWe prove that for every integer k, every finite set of points in the plane can be k-colored ...
The art gallery problem asks for the smallest number of guards required to see every point of the in...
Several important general theorems of Euclidean Ramsey Theory are presented with an emphasis on tryi...
Let S be a two-colored set of n points in general position in the plane. We show that S admits at l...
Improving a result of Aichholzer et al., we show that there exists a constant c > 0 satisfying the f...
AbstractGiven a set T of nonnegative integers, a T-coloring of a graph G is a labeling of the vertic...
A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignme...
To a set of n points in the plane, one can associate a graph that has less than n 2 vertices and has...
We prove that for any point set P in the plane, a triangle T, and a positive integer k, there exists...
Abstract. We prove that for any finite point set P in the plane, a triangle T, and a positive intege...
International audienceWe give new positive results on the long-standing open problem of geometric co...
We prove that every finite set of homothetic copies of a given convex body in the plane can be color...
Our point of departure is the following simple common generalisation of the Sylvester–Gallai theorem...
AbstractThe 512 points of an 8 × 8 × 8 cubical array of lattice points in 3-space can be colored in ...
Let <em>P</em> be a set of <em>n</em> points in general position in the plane. We study the chromati...
AbstractWe prove that for every integer k, every finite set of points in the plane can be k-colored ...
The art gallery problem asks for the smallest number of guards required to see every point of the in...
Several important general theorems of Euclidean Ramsey Theory are presented with an emphasis on tryi...
Let S be a two-colored set of n points in general position in the plane. We show that S admits at l...
Improving a result of Aichholzer et al., we show that there exists a constant c > 0 satisfying the f...
AbstractGiven a set T of nonnegative integers, a T-coloring of a graph G is a labeling of the vertic...
A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignme...
To a set of n points in the plane, one can associate a graph that has less than n 2 vertices and has...