Families of vectors without antipodal pairs

Some Erd\H{o}s-Ko-Rado type extremal properties of families of vectors from $\{-1,0,1\}^n$ are considered

Intersection theorems for (−1,0,1)(-1,0,1)-vectors

In this paper, we investigate Erd\H os--Ko--Rado type theorems for families of vectors from $\{0,\pm 1\}^n$ with fixed numbers of $+1$'s and $-1$'s. Scalar product plays the role of intersection size. In particular, we sharpen our earlier result on the largest size of a family of such vectors that avoids the smallest possible scalar product. We also obtain an exact result for the largest size of a family with no negative scalar products

Binary scalar products

Let $A,B \subseteq \mathbb{R}^d$ both span $\mathbb{R}^d$ such that $\langle a, b \rangle \in \{0,1\}$ holds for all $a \in A$, $b \in B$. We show that $|A| \cdot |B| \le (d+1) 2^d$. This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) concerning 2-level polytopes. Such polytopes have the property that for every facet-defining hyperplane $H$ there is a parallel hyperplane $H'$ such that $H \cup H'$ contain all vertices. The authors conjectured that for every $d$-dimensional 2-level polytope $P$ the product of the number of vertices of $P$ and the number of facets of $P$ is at most $d 2^{d+1}$, which we show to be true.Comment: 10 page

Independence numbers of Johnson-type graphs

We consider a family of distance graphs in $\mathbb{R}^n$ and find its independent numbers in some cases. Define graph $J_{\pm}(n,k,t)$ in the following way: the vertex set consists of all vectors from $\{-1,0,1\}^n$ with $k$ nonzero coordinates; edges connect the pairs of vertices with scalar product $t$. We find the independence number of $J_{\pm}(n,k,t)$ for $n > n_0 (k,t)$ in the cases $t = 0$ and $t = -1$; these cases for $k = 3$ are solved completely. Also the independence number is found for negative odd $t$ and $n > n_0 (k,t)$

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend