Related papers: On Triangles in Colored Pseudoline Arrangements
A graph is pseudo-outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, the…
It is consistent that ZF+DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles in two-dimensional Euclidean space does not.
In 1973 P. Erd\H{o}s and L. Lov\'asz noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erd\H{o}s and Lov\'asz proved that…
We prove that any finite set of half-planes can be colored by two colors so that every point of the plane, which belongs to at least three half-planes in the set, is covered by half-planes of both colors. This settles a problem of Keszegh.
A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. If the coloring is required to be proper, then the upper bound for the…
Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all 3 colours. What happens for more colours: if we…
We study thresholds for the appearance of a 2-core in random hypergraphs that are a mixture of a constant number of random uniform hypergraphs each with a linear number of edges but with different edge sizes. For the case of two overlapping…
A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Gr\"unbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two…
We consider arrangements of $n$ pseudo-lines in the Euclidean plane where each pseudo-line $\ell_i$ is represented by a bi-infinite connected $x$-monotone curve $f_i(x)$, $x \in \mathbb{R}$, s.t.\ for any two pseudo-lines $\ell_i$ and…
A coloring of the $\ell$-dimensional faces of $Q_n$ is called $d$-polychromatic if every embedded $Q_d$ has every color on at least one face. Denote by $p^\ell(d)$ the maximum number of colors such that any $Q_n$ can be colored in this way.…
We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace…
The precoloring problem of a graph involves assigning colors to some vertices beforehand, and the objective is to determine whether it can be extended to a proper k-coloring of the entire graph. In 1958, Grotzsch proved that every…
Stan Wagon asked the following in 2000. Is every zonohedron face 3-colorable when viewed as a planar map? An equivalent question, under a different guise, is the following: is the arrangement graph of great circles on the sphere always…
In this paper we study threshold coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is…
A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. Fabrici and G\"oring proved that six colors are…
Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to…
We use Menger's Theorem and K\"onig's Line Colouring Theorem to show that in any tripartite graph with two complete (bipartite) sides the maximum number of pairwise edge-disjoint triangles equals the minimum number of edges that meet all…
In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings…
Let $H_{n,(p_m)_{m=2,\ldots,M}}$ be a random non-uniform hypergraph of dimension $M$ on $2n$ vertices, where the vertices are split into two disjoint sets of size $n$, and colored by two distinct colors. Each non-monochromatic edge of size…
We discuss representations and colorings of orthogonality hypergraphs in terms of their two-valued states interpretable as classical truth assignments. Such hypergraphs, if they allow for a faithful orthogonal representation, have quantum…