Related papers: Coloring triangles and rectangles
It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles on a given Euclidean space has countable chromatic number, while the hypergraph of isosceles triangles in the plane does not.
It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles in Euclidean plane has countable chromatic number, while there is no Vitali set.
Let n>1 be a number. Let Gn be the hypergraph of all rectangles in an n-dimensional Euclidean space. It is consistent that ZF+DC holds, the chromatic number of Gn is countable, yet the chromatic number of Gn+1 is uncountable.
Let Gn be the graph on n-dimensional Euclidean space connecting points of rational Euclidean distance. It is consistent relative to an inaccessible cardinal that ZF+DC holds and G3 has countable chromatic number, yet G4 has uncountable…
Let n>0 be a number. Let Gn be the graph on n-dimensional Euclidean space connecting points of rational distance. It is consistent with the choiceless theory ZF+DC that Gn has countable chromatic number yet Gn+1 does not.
We prove several consistency results in choiceless set theory ZF+DC regarding countable chromatic numbers of various algebraic hypergraphs on Euclidean spaces.
If the chromatic number of Euclidean plane is larger than four, but it is known that the chromatic number of planar graphs is equal to four, then how does one explain it? In my opinion, they are contradictory to each other. This idea leads…
In the paper we prove, in particular, that for any measurable coloring of the euclidian plane into two colours there is a monochromatic triangle with some restrictions on the sides. Also we consider similar problems in finite fields…
The chromatic number of a subset of Euclidean space is the minimal number of colors sufficient for coloring all points of this subset in such a way that any two points at the distance 1 have different colors. We give new upper bounds for…
We consider the faces in pseudoline arrangements in which the pseudolines are colored with two colors. Bj\"orner, Las Vergnas, Sturmfels, White, and Ziegler conjecture the existence of a two-colored triangle in such arrangements. We…
A k-uniform hypergraph is algebraic if its vertex set is n-dimensional Euclidean space, for some n, and its hyperedge set is defined from the zero set of some polynomial. The chromatic numbers of all algebraic hypergraphs are determined,…
We consider the polychromatic coloring problems for unions of two or more geometric hypergraphs on the same vertex sets of points in the plane. We show, inter alia, that the union of bottomless rectangles and horizontal strips does in…
The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but…
This is a treatise on finite point configurations spanning a fixed volume to be found in a single color-class of an arbitrary finite (measurable) coloring of the Euclidean space $\mathbb{R}^n$, or in a single large measurable subset…
We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this…
The \emph{chromatic number} of a hypergraph is the smallest number of colors needed to color the vertices such that no edge of at least two vertices is monochromatic. Given a family of geometric objects $\mathcal{F}$ that covers a subset…
Recently, the homology and cohomology of non-k-overlapping discs, or, equivalently, no k-equal subspaces of Euclidean space, were calculated by Dobrinskaya and Turchin. We calculate the homology and cohomology of two classes of more general…
We popularize the question whether, for $m$ large enough, all $m$-uniform shift-chain hypergraphs are properly $2$-colorable. On the other hand, we show that for every $m$ some $m$-uniform shift-chains are not polychromatic $3$-colorable.
Let $G$ be the unit distance graph in the plane. A well-known problem in combinatorial geometry is that of determining the chromatic number of $G$. It is known that $4\le \chi(G)\le 7$. The upper bound of 7 is obtained using tilings of the…
Motivated by an old conjecture of P. Erd\H{o}s and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable…