Related papers: A note on the colorful fractional Helly theorem

A graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. Motivated by previous work on dually chordal graphs and graphs of bounded distance VC-dimension we prove several new results on the…

We prove the following colorful Helly-type result: Fix $k \in [d-1]$. Assume $\mathcal{A}_1, \dots, \mathcal{A}_{d+(d-k)+1}$ are finite sets (colors) of nonzero vectors in $\R^d$. If for every rainbow sub-selection $R$ from these sets of…

Hadwiger's transversal theorem gives necessary and sufficient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful…

The 1913 Helly's theorem states that any family ${\cal K}$ of $n\geq d+1$ convex sets in ${\mathbb R}^d$ can be pierced by a single point if and only if any $d+1$ of ${\cal K}$'s elements can. In 2002 Alon, Kalai, Matou\v{s}ek and Meshulam…

Given a graph $G$ and a collection $\mathcal C$ of subsets of $\mathbb{R}^d$ indexed by the subsets of vertices of $G$, a constrained drawing of $G$ is a drawing, where each edge is drawn inside some set from $\mathcal C$, in such a way…

We propose an interpretation of, and approach to, Helly's theorem that can be included quite early in the undergraduate curriculum. At the same time, the approach connects with contemporary models of data privacy and with sampling methods…

We prove that two closed subsets of complex space $\C^n$ with corresponding complex homothetic sections (projections) are complex homothetic. The proof uses a new Helly-type theorem for cosets of closed subgroups of $\S ^1$.

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…

We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous…

We prove the following variant of Helly's classical theorem for Hamming balls with a bounded radius. For $n>t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X^n$, every subfamily of at most…

We establish a colorful and, more generally, matroidal solution to the problem of Goodman and Pollack on the existence of an $\mathbb{F}$-affine $k$-dimensional transversal to a family of convex sets in $\mathbb{F}^d$, where $0 \le k \le d…

We give a Belyi-type characterisation of smooth complete intersections of general type over $\mathbb{C}$ which can be defined over $\bar{\mathbb{Q}}$. Our proof uses the higher-dimensional analogue of the Shafarevich boundedness conjecture…

Brazitikos' results on quantititative Helly-type theorems (for the volume and for the diameter) rely on the work of Srivastava on sparsification of John's decompositions. We change this technique by a stronger recent result due to Friedland…

The coloured Tverberg theorem was conjectured by B\'ar\'any, Lov\'{a}sz and F\"uredi and asks whether for any d+1 sets (considered as colour classes) of k points each in R^d there is a partition of them into k colourful sets whose convex…

Hall's Theorem is a basic result in Combinatorics which states that the obvious necesssary condition for a finite family of sets to have a transversal is also sufficient. We present a sufficient (but not necessary) condition on the sizes of…

We prove a no-dimensional Helly theorem for affine spaces and convex sets using the unboundedness framework of Aronov, Goodman, and Pollack (Computational Geometry, 2002). This generalizes the fundamental result of Adiprasito, B\'ar\'any,…

The Bell numbers count the number of different ways to partition a set of $n$ elements while the graphical Bell numbers count the number of non-equivalent partitions of the vertex set of a graph into stable sets. This relation between graph…

Let $G$ be a simple graph with $n$ vertices and list chromatic number $\chi_\ell(G)=\chi_\ell$. Suppose that $0\leq t\leq \chi_\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Albertson, Grossman and Haas [1] conjectured that…

Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…

We investigate the complexity of generalizations of colourings (acyclic colourings, $(k,\ell)$-colourings, homomorphisms, and matrix partitions), for the class of transitive digraphs. Even though transitive digraphs are nicely structured,…