Related papers: Helly Numbers of Polyominoes

In this paper we study generalizations of classical results on intersection patterns of set systems in $\mathbb{R}^d$, such as the fractional Helly theorem or the $(p,q)$-theorem, in the setting of arbitrary triangulable spaces with a…

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where…

We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in $\mathbb{R}^d$. Halman's theorem says that, given a set $S$ in $\mathbb{R}^d$, if $F$ is a finite family of axis-parallel boxes such that the intersection of…

Let Y be a random variable satisfying specific moment conditions. This paper introduces and investigates probabilistic heterogeneous Stirling numbers of the second kind and probabilistic heterogeneous Bell polynomials. These structures…

Let $p$ be a prime number and $k$ be a positive integer not divisible by $p$. We describe the Heller translates of the periodic Lie module $\mathrm{Lie}(pk)$ in characteristic $p$ and show that it has period $2p-2$ when $p$ is odd and $1$…

Given a clover link, we construct a bottom tangle by using a disk/band surface of the clover link. Since the Milnor number is already defined for a bottom tangle, we define the Milnor number for the clover link to be the Milnor number for…

Belyi's Theorem states that a Riemann surface, X, as an algebraic curve is defined over an algebraic closure of the rationals if and only if there exists a holomorphic function taking X to the Riemann sphere with at most three critical…

Let $n\geq 1$, $0\leq t\leq {n \choose 2}$ be arbitrary integers. Define the numbers $I_n(t)$ as the number of permutations of $[n]$ with $t$ inversions. Let $n,d\geq 1$ and $0\leq t\leq (d-1)n$ be arbitrary integers. Define {\em the…

In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new…

We provide a number of new quantitative versions of Helly's theorem. For example, we show that for every family $\{P_i:i\in I\}$ of closed half-spaces $$P_i=\{x\in {\mathbb R}^n:\langle x,w_i\rangle \leq 1\}$$ in ${\mathbb R}^n$ such that…

Let $ \{P_n\}_{n\geq 0} $ be the sequence of Perrin numbers defined by $P_0=3$, $P_1=0$,$P_2=2$ and $P_{n+3}=P_{n+1}+P_{n}$ for all $n \geq 0$. In this paper, we determine all Perrin numbers that are palindromic concatenations of two…

A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of…

We classify path polyominoes which are level and pseudo-Gorenstein. Moreover, we compute all level and pseudo-Gorenstein simple thin polyominoes with rank less than or equal to 10. We also compute the regularity of the pseudo-Gorenstein…

The height of an algebraic number $\alpha$ is a measure of how arithmetically complicated $\alpha$ is. We say $\alpha$ is totally $p$-adic if the minimal polynomial of $\alpha$ splits completely over the field $\mathbb{Q}_p$ of $p$-adic…

We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid -- the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of…

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This…

Here, we show that there is no positive integer $n$ such that the $n$th Cullen number $C_n=n2^n+1$ has the property that it is composite but $\phi(C_n)\mid C_n-1$.

The edge intersection graph of a family of paths in host tree is called an $EPT$ graph. When the host tree has maximum degree $h$, we say that $G$ belongs to the class $[h,2,2]$. If, in addition, the family of paths satisfies the Helly…

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,…

Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a…