Related papers: A combinatorial approach to colourful simplicial d…

We study the Orchard relation for generic configurations of points in the plane (also called order types). We introduce infinitesimally-close points and analyse the relation of this notion with the Orchard relation. The second part of the…

An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^2$, is a partition of $\mathbb{S}^2$ into parts such that no part contains a pair of orthogonal points, that is, a pair of points at spherical distance $\pi/2$ apart. It…

A particular discrete rhombohedral lattice consisting of four symmetrically interlaced cuboctahedral point lattices is described that is interesting because of the high degree of symmetry it exhibits. The four constituent cuboctahedral…

Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours.…

We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree…

Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in…

We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral…

We define an (r,s)-coloring of an abstract simplicial complex to be a coloring using r colors of the vertices so that in any simplex at most s vertices have the same color. We translate the problem of finding an (r,s)-coloring of a given…

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an n-point set $P$, and a collection $\mathcal{F} =…

Higher chromatic numbers $\chi_s$ of simplicial complexes naturally generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily large for…

Let $P$ be a set of $n$ points in $d$-dimensions. The simplicial depth, $\sigma_P(q)$ of a point $q$ is the number of $d$-simplices with vertices in $P$ that contain $q$ in their convex hulls. The simplicial depth is a notion of data depth…

Let $X_1,\ldots, X_{d+2}$ be random points in $\mathbb R^d$. The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by $P:= [X_1,\ldots, X_{d+2}]$, is a simplex. In the present paper,…

A vertex coloring of a simplicial complex $\Delta$ is called a linear coloring if it satisfies the property that for every pair of facets $(F_1, F_2)$ of $\Delta$, there exists no pair of vertices $(v_1, v_2)$ with the same color such that…

In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use…

Let $\mathcal{F}$ be a family of convex sets in ${\mathbb R}^d$, which are colored with $d+1$ colors. We say that $\mathcal{F}$ satisfies the Colorful Helly Property if every rainbow selection of $d+1$ sets, one set from each color class,…

The inclusion relation between simple objects in the plane may be used to define geometric set systems, or hypergraphs. Properties of various types of colorings of these hypergraphs have been the subject of recent investigations, with…

Carath\'eodorys Theorem of convex hulls plays an important role in convex geometry. In 1982, B\'ar\'any formulated and proved a more general version, called the Colorful Carath\'eodory. This colorful version was even more generalized by…

The colorful Carath\'eodory theorem, proved by B\'ar\'any in 1982, states that given d+1 sets of points S_1,...,S_{d+1} in R^d, with each S_i containing 0 in its convex hull, there exists a subset T of the union of the S_i's containing 0 in…

Tverberg's theorem states that for any $k \ge 2$ and any set $P \subset \mathbb{R}^d$ of at least $(d + 1)(k - 1) + 1$ points in $d$ dimensions, we can partition $P$ into $k$ subsets whose convex hulls have a non-empty intersection. The…

Youngs proved that every non-bipartite quadrangulation of the projective plane $\mathbb{R}\mathrm{P}^2$ is 4-chromatic. Kaiser and Stehl\'{\i}k [J. Combin. Theory Ser. B 113 (2015), 1-17] generalised the notion of a quadrangulation to…