Related papers: A 920-block explicit construction guaranteeing a t…

We present a simple explicit family $\mathcal{B}$ of $910$ $6$-subsets of $[60]=\{1,\dots,60\}$ such that every $6$-subset $S\subset[60]$ intersects at least one block $B\in\mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$.…

We exhibit an explicit family $\mathcal{B}$ of $30$ subsets (``blocks'') of size $6$ of $[60]=\{1,2,\dots,60\}$ with the following property: for every $6$-subset $S\subset[60]$, there exists a block $B\in\mathcal{B}$ such that $|S\cap B|\ge…

In this paper, six constructions of difference families are presented. These constructions make use of difference sets, almost difference sets and disjoint difference families, and give new point of views of relationships among these…

A family $\mathcal{F}$ on ground set $\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this…

It is demonstrated that each nearly neighbourly family of standard boxes in $\mathbb{R}^3$ has at most 12 elements. A combinatorial classification of all such families that have exactly 12 elements is given. All families satisfying an extra…

Let $\mathcal F\subset 2^{[n]}$ be a family in which any three sets have non-empty intersection and any two sets have at least $38$ elements in common. The nearly best possible bound $|\mathcal F|\le 2^{n-2}$ is proved. We believe that $38$…

A family $\mathcal{F}$ on ground set $[n]:=\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of at most $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while…

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the…

We construct, for any positive integer n, a family of n congruent convex polyhedra in R^3, such that every pair intersects in a common facet. Previously, the largest such family contained only eight polytopes. Our polyhedra are Voronoi…

We study a construction, which produces surfaces $Y \subset P_3$ with cusps. For example we obtain surfaces of degree six with 18, 24 or 27 three-divisible cusps. For sextic surfaces in a particular family of up to 30 cusps the codes of…

A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results…

For a simplicial complex ${\mathcal C}$ denote by $\beta({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…

Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$,…

We prove three theorems giving extremal bounds on the incidence structures determined by subsets of the points and blocks of a balanced incomplete block design (BIBD). These results generalize and strengthen known bounds on the number of…

We introduce near triple arrays as binary row-column designs with at most two consecutive values for the replication numbers of symbols, for the intersection sizes of pairs of rows, pairs of columns and pairs of a row and a column. Near…

Let $A_{3}(n)$ (resp. ${{B}_{3}}(n)$) denote the number of partition pairs (resp. triples) of $n$ where each partition is 3-core. By applying Ramanujan's ${}_{1}\psi_{1}$ formula and Bailey's ${}_{6}\psi_{6}$ formula, we find the explicit…

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of…

For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in…

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets $A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block…