Related papers: Extensions of discrete Helly theorems for boxes

Hindman's Theorem (HT) states that for every coloring of $\mathbb N$ with finitely many colors, there is an infinite set $H \subseteq \mathbb N$ such that all nonempty sums of distinct elements of $H$ have the same color. The investigation…

Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several…

A particular case of the Hindman--Galvin--Glazer theorem states that, for every partition of an infinite abelian group $G$ into two cells, there will be an infinite $X\subseteq G$ such that the set of its finite sums…

Let $K_d$ denote the class of all finite graphs and, for graphs $A\subseteq B$, say $A \leq_d B$ if distances in $A$ are preserved in $B$; i.e. for $a, a' \in A$ the length of the shortest path in $A$ from $a$ to $a'$ is the same as the…

We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every…

Cohen et al. conjectured that for every oriented cycle $C$ there exist an integer $f(C)$ such that every strong $f(C)$-chromatic digraph contains a subdivision of $C$. El Joubbeh confirmed this conjecture for Hamiltonian digraphs. Indeed,…

A family of axis-aligned boxes in $\er^d$ is \emph{$k$-neighborly} if the intersection of every two of them has dimension at least $d-k$ and at most $d-1$. Let $n(k,d)$ denote the maximum size of such a family. It is known that $n(k,d)$ can…

Hindman's theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. Ramsey algebras are structures that satisfy an analogue of Hindman's Theorem. This paper introduces Ramsey algebras and…

In this paper we examine the reverse mathematical strength of a variation of Hindman's Theorem HT constructed by essentially combining HT with the Thin Set Theorem TS to obtain a principle which we call thin-HT. thin-HT says that every…

We study the existence of fixed points for continuous maps $f$ from an $n$-ball $X$ in $\mathbb R^n$ to $\mathbb R^n$ with $n\geq 1$. We show that $f$ has a fixed point if, for some absolute retract $Y\subset\partial X$, $f(Y)\subset X$ and…

A basic measure of the combinatorial complexity of a convexity space is its Radon number. In this paper we show a fractional Helly theorem for convexity spaces with a bounded Radon number, answering a question of Kalai. As a consequence we…

We show that for every infinite collection $\mathcal{R}$ of disjoint equivalent rays in a graph $G$ there is a subdivision of the hexagonal half-grid in $G$ such that all its vertical rays belong to $\mathcal{R}$. This result strengthens…

Let $A$ be a finite, abelian group. We show that the density of $A$-extensions satisfying the Hasse norm principle exists, when the extensions are ordered by discriminant. This strengthens earlier work of Frei--Loughran--Newton \cite{FLN},…

Suppose that $nk$ points in general position in the plane are colored red and blue, with at least $n$ points of each color. We show that then there exist $n$ pairwise disjoint convex sets, each of them containing $k$ of the points, and each…

We show that if A is a large subset of a box in Z^d with dimensions L_1 >= L_2 >= ... >= L_d which are all reasonably large, then |A + A| > 2^{d/48}|A|. By combining this with Chang's quantitative version of Freiman's theorem, we prove a…

We show that for any compact convex set $K$ in $\mathbb{R}^d$ and any finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$, if the intersection of every sufficiently small subfamily of $\mathcal{F}$ contains an isometric copy of $K$…

For an integer $d \geq 2$, a family $\mathcal{F}$ of sets is $\textit{$d$-wise intersecting}$ if for any distinct sets $A_1,A_2,\dots,A_d \in \mathcal{F}$, $A_1 \cap A_2 \cap \dots \cap A_d \neq \emptyset$, and $\textit{non-trivial}$ if…

A result of Helson on general Dirichlet series $\sum a_{n} e^{-\lambda_{n}s}$ states that, whenever $(a_{n})$ is $2$-summable and $\lambda=(\lambda_{n})$ satisfies a certain condition introduced by Bohr, then for almost all homomorphism…

Carath\'eodory's, Helly's and Radon's theorems are three basic results in discrete geometry. Their max-plus counterparts have been proved by various authors. In this paper, more advanced results in discrete geometry are shown to have also…

Helly's selection theorem provides a criterion for compactness of sets of single-variable functions with bounded pointwise variation. Fra{\v{n}}kov{\'a} has given a proper extension of Helly's theorem to the setting of single-variable…