Related papers: Non-trivial $t$-intersecting families for vector s…

We show that an $n$-uniform maximal intersecting family has size at most $e^{-n^{0.5+o(1)}}n^n$. This improves a recent bound by Frankl. The Spread Lemma of Alweiss, Lowett, Wu and Zhang plays an important role in the proof.

Let $V$ be a real or complex vector space. The finite topology of $V$ consists of all the subsets $U$ for which the intersection $U \cap F$ is closed in $F$ for every finite-dimensional linear subspace of $V$. It is known that if $V$ has…

We construct an intersecting $k$-family of transversal size $\lceil \frac{k+1}{2} \rceil$ and length $k+1$ and study some of its properties. We use this family to prove that $q(4) = 9$. We also construct a $k$-family for $k = 2^m - 1$ of…

One way to study the combinatorics of finite metric spaces is to study the betweenness relation associated with the metric space. In the hypergraph metrization problem, one has to find and characterize metric betweennesses whose collinear…

Let $G$ be a finite group. A family $\mathcal{M}$ of maximal subgroups of $G$ is called `irredundant' if its intersection is not equal to the intersection of any proper subfamily. $\mathcal{M}$ is called `maximal irredundant' if…

We continue the study by Melo and Winter [arXiv:1712.01763, 2017] on the possible intersection sizes of a $k$-dimensional subspace with the vertices of the $n$-dimensional hypercube in Euclidean space. Melo and Winter conjectured that all…

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…

Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in 3-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal…

Let $S$ be a set of $n$ points in the plane in general position. Two line segments connecting pairs of points of $S$ cross if they have an interior point in common. Two vertex disjoint geometric graphs with vertices in $S$ cross if there…

We prove that the maximum size of a family of $k$-element subsets of the set $[n] = \{1, 2, \ldots, n\}$ which contains no singleton intersection is $\binom{n-2}{k-2}$ when $3k-3 \le n \le k^2-k+1$. This improves upon a recent result of…

Let $V$ denote an $r$-dimensional vector space over $\mathbb{F}_{q^n}$, the finite field of $q^n$ elements. Then $V$ is also an $rn$-dimension vector space over $\mathbb{F}_q$. An $\mathbb{F}_q$-subspace $U$ of $V$ is $(h,k)_q$-evasive if…

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…

For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2, \ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting} iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing a question…

We study the following natural arithmetic question regarding intersecting families: how large can a family of subsets of integers from $\{1, \ldots n\}$ be such that, for every pair of subsets in the family, the intersection contains a sum…

A bounded linear operator T on Hilbert space is subspace-hypercyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace in a relatively dense set. We construct examples to show that subspace-hypercyclicity…

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $\left[V\atop k\right]_q$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ is called…

The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In…

A family of subsets of the set {1,2,...,n} is said to be unbalanced if the convex hull of its characteristic vectors misses the diagonal in the n-cube.The purpose of this article is to develop the combinatorics of maximal unbalanced…

Let F be a field. We investigate the greatest possible dimension t_n(F) for a vector space of n-by-n matrices with entries in F and in which every element is triangularizable over the ground field F. It is obvious that t_n(F) is greater…

Let $t$, $r$, $k$ and $n$ be positive integers and $\mathcal{F}$ a family of $k$-subsets of an $n$-set $V$. The family $ \CF $ is $ r $-wise $ t $-intersecting if for any $ F_1, \ldots, F_r \in \CF $, we have $ \abs{\cap_{i = 1}^{r}F_i}\gs…