Related papers: Generalized laminar families and certain forbidden…

We prove that if the prime graphs in a graph class have bounded lettericity, then the entire class has bounded lettericity if and only if it does not contain arbitrary large matchings, co-matchings, or a family of graphs that we call…

Let $E\supseteq F$ be a field extension and $M$ a graded Lie algebra of maximal class over $E$. We investigate the $F$-subalgebras $L$ of $M$, generated by elements of degree $1$. We provide conditions for $L$ being either ideally…

It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a…

A set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ is called $\textit{bilinear}$ when it is the zero set of a family of linear and bilinear forms, and $\textit{transverse}$ when it is stable under vertical and horizontal sums. A theorem of…

A family $\mathcal{F}$ is $t$-$\it{intersecting}$ if any two members have at least $t$ common elements. Erd\H os, Ko, and Rado proved that the maximum size of a $t$-intersecting family of subsets of size $k$ is equal to $ {{n-t} \choose…

Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F: X \longrightarrow B_{X}$ be a set-valued mapping such that $x \in F(x)$, $sup_{x\in X} | F(x)|< \kappa$, $sup_{x\in X} | F^{-1}(x)|< \kappa$ for all $x\in X$ and…

A family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ is called a $t$-intersecting family if $|F\cap G| \geq t$ for any two members $F, G \in \mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\mathcal{F}$…

A set of sets is called a family. 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 $1 \leq…

A family $\mathcal{F}$ of subsets of $[n]=\{1,2,\ldots,n\}$ shatters a set $A \subseteq [n]$ if for every $A' \subseteq A$ there is an $F \in \mathcal{F}$ such that $F \cap A=A'$. We develop a framework to analyze $f(n,k,d)$, the maximum…

A graph is said to be orthogonalisable if the set of real symmetric matrices whose off-diagonal pattern is prescribed by its edges contains an orthogonal matrix. We determine some necessary and some sufficient conditions on the sizes of the…

A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically…

Consider a family $\mathcal{F}$ of $k$-subsets of an ambient $(k^2-k+1)$-set such that no pair of $k$-subsets in $\mathcal{F}$ intersects in exactly one element. In this short note we show that the maximal size of such $\mathcal{F}$ is…

Given a poset $P$, a family $F$ of elements in the Boolean lattice is said to be $P$-saturated if (1) $F$ contains no copy of $P$ as a subposet and (2) every proper superset of $F$ contains a copy of $P$ as a subposet. The maximum size of a…

The family, L(INDLIN), of languages generated by linear indexed grammars has been studied in the literature. It is known that the Parikh image of every language in L(INDLIN) is semi-linear. However, there are bounded semi linear languages…

We study families of positive and completely positive maps acting on a bipartite system $\mathbb{C}^M\otimes \mathbb{C}^N$ (with $M\leq N$). The maps have a property that when applied to any state (of a given entanglement class) they result…

Let $\Gamma$ be an abelian group and $g \geq h \geq 2$ be integers. A set $A \subset \Gamma$ is a $C_h[g]$-set if given any set $X \subset \Gamma$ with $|X| = k$, and any set $\{ k_1 , \dots , k_g \} \subset \Gamma$, at least one of the…

We say that a set $A$ \emph{$t$-intersects} a set $B$ if $A$ and $B$ have at least $t$ common elements. Two families $\mathcal{A}$ and $\mathcal{B}$ are said to be \emph{cross-$t$-intersecting} if each set in $\mathcal{A}$ $t$-intersects…

Chv\'{a}tal conjectured that a star is amongst the largest intersecting subfamiles of a finite subset-closed family of sets. Kleitman later strengthened Chv\'{a}tal's conjecture, suggesting that maximal intersecting subfamilies of $2^{[n]}$…

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 $A\in\mathcal{A}$,…

Let $G$ be a unipotent group and $\mathcal F=\{F_t:t\in (0,\infty)\}$ a family of subsets of $G$, with $\mathcal F$ definable in an o-minimal expansion of the real field. Given a lattice $\Gamma\subseteq G$, we study the possible Hausdorff…