Related papers: Small doubling, atomic structure and $\ell$-divisi…

The study of `structure' on subsets of abelian groups, with small `doubling constant', has been well studied in the last fifty years, from the time Freiman initiated the subject. In \cite{DF} Deshouillers and Freiman establish a structure…

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of…

We show that if $B \subset \mathbb{R}^n$ and $E \subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $\mathbb{R}^n$ such that every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least $\alpha$ with…

A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Families $\mathcal{A}_1, \mathcal{A}_2, \dots, \mathcal{A}_k$ of sets are cross-$t$-intersecting if, for every $i$ and $j$ in $\{1, 2, \dots, k\}$ with $i…

Let $k, r, n \geq 1$ be integers, and let $\S_{n, k, r}$ be the family of $r$-signed $k$-sets on $[n] = \{1, \dots, n\}$ given by $$ \mathcal{S}_{n, k, r} = \Big\{\{(x_1, a_1), \dots, (x_k, a_k)\}: \{x_1, \dots, x_k\} \in \binom{[n]}{k},…

A family of perfect matchings of $K_{2n}$ is $intersecting$ if any two of its members have an edge in common. It is known that if $\mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, then $|\mathcal{F}| \leq (2n-3)!!$ and…

Paul Erd\H{o}s and L\'aszl\'o Lov\'asz established that any \emph{maximal intersecting family of $k-$sets} has at most $k^{k}$ blocks. They introduced the problem of finding the maximum possible number of blocks in such a family. They also…

We study $t$-intersecting and $t$-cross-intersecting families of $k$-dimensional subspaces in finite vector spaces of dimension $n$. We show that all large $t$-intersecting families admit a governing low-dimensional structure for $n \ge…

A family $\F$ of sets is said to be intersecting if any two sets in $\F$ have nonempty intersection. The celebrated Erd{\H o}s-Ko-Rado theorem determines the size and structure of the largest intersecting family of $k$-sets on an $n$-set…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. Let $f(n,a)$ be the maximum number of sets in a…

We say a family of sets is intersecting if any two of its sets intersect, and we say it is trivially intersecting if there is an element which appears in every set of the family. In this paper we study the maximum size of a non-trivially…

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called {\it non-trivial cross-intersecting} if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in…

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a set $S\subseteq [n]$ if every possible subset of $S$ appears as the intersection of $S$ with some element of $\mathcal{F}$ and we denote by $\text{Sh}(\mathcal{F})$ the…

Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets…

A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family $\mathcal{F}\subseteq \mathcal{P}(n)$ that does not contain a $2$-chain $F_1\subsetneq F_2$. Erd\H{o}s later extended this result and…

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\cF$ is a family of subsets of $X$ such that no two sets $A, B \in \cF$ satisfy $A \subset B$ (or $B \subset…

A collection of $k$ sets is said to form a $k$-sunflower, or $\Delta$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets $\mathcal{F}$ sunflower-free if it contains no sunflowers.…

More than 50 years ago, Erd\H os asked the following question: what is the maximum size of a family $\mathcal F$ of $k$-element subsets of an $n$-element set if it has no $s+1$ pairwise disjoint sets? This question attracted a lot of…

For integers $n\ge s\ge2$, let $e(n,s)$ be the maximum size of a family $\mathcal F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The study of determining $e(n,s)$ is closely related to its uniform counterpart, the well-known…

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$…