Related papers: A fractional Helly theorem for boxes

A subset $A$ of $[n] = \{1, \dots, n\}$ is $k$-separated if, when the elements of $[n]$ are considered on a circle, between any two elements of $A$ there are at least $k$ elements of $[n]$ that are not in $A$. A family $\mathcal{A}$ of sets…

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon}…

Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$, and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be…

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group…

We show that there is some absolute constant $c>0$, such that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}$, if \mbox{$|\mathcal{F}| \geq (\frac{1}{2}-c)2^n$}, then there is some element $i \in [n]$ that appears in at least…

We prove that for every $\epsilon>0$ there exists $\delta>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-\epsilon)\frac{n^{2}}{2}$ pairs of curves…

For $\alpha>1$ we represent a real number in $(0,1]$ in the form \[ \sum_{i=1}^{\infty}(\alpha-1)^{i-1}\alpha^{-(d_{1}+\dots+d_{i})}\] with $d_{i}\in\mathbb{N}$. We discuss ergodic theoretical and dimension theoretical aspects of this…

Let $M$ be a simple 3-manifold, and $F$ be a component of $\partial M$ of genus at least 2. Let $\alpha$ and $\beta$ be separating slopes on $F$. Let $M(\alpha)$ (resp. $M(\beta)$) be the manifold obtained by adding a 2-handle along…

A family of subsets $\mathcal{F}$ is intersecting if $A \cap B \neq \emptyset$ for any $A, B \in \mathcal{F}$. In this paper, we show that for given integers $k > d \ge 2$ and $n \ge 2k+2d-3$, and any intersecting family $\mathcal{F}$ of…

It is well-known that a line can intersect at most $2n-1$ cells of the $n \times n$ chessboard. Here we consider the high dimensional version: how many cells of the $d$-dimensional $n\times \ldots \times n$ box can a hyperplane intersect?…

In this paper we deal with a problem of Peth\H{o} related to existence of quartic algebraic integer $\alpha$ for which $$ \beta=\frac{4\alpha^4}{\alpha^4-1}-\frac{\alpha}{\alpha-1} $$ is a quadratic algebraic number. By studying rational…

A family $\mbox{$\cal F$}=\{F_1,\ldots,F_m\}$ of subsets of $[n]$ is said to be ordered, if there exists an $1\leq r\leq m$ index such that $n\in F_i$ for each $1\leq i\leq r$, $n\notin F_i$ for each $i>r$ and $|F_i|\leq |F_j|$ for each…

Let $H$ be a complete $r$-uniform hypergraph such that two vertices are marked in each edge as its `boundary' vertices. A linear ordering of the vertex set of $H$ is called an {\em agreeing linear order}, provided all vertices of each edge…

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the…

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost…

A family F is intersecting if any two members have a nonempty intersection. Erdos, Ko, and Rado showed that |F|\leq {n-1\choose k-1} holds for an intersecting family of k-subsets of [n]:={1,2,3,...,n}, n\geq 2k. For n> 2k the only extremal…

We show that if $-A$ generates a bounded $\alpha$-times resolvent family for some $\alpha \in (0,2]$, then $-A^{\beta}$ generates an analytic $\gamma$-times resolvent family for $\beta \in(0,\frac{2\pi-\pi\gamma}{2\pi-\pi\alpha})$ and…

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is…

Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. For $\alpha \in \mathcal{O}_K$ with $|\alpha| > 1$, define \[ \mathcal{D}_\alpha = \bigcup_{n=0}^\infty…

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…