Related papers: Intersective sets for sparse sets of integers

Intersective polynomials are polynomials in $\Z[x]$ having roots every modulus. For example, $P_1(n)=n^2$ and $P_2(n)=n^2-1$ are intersective polynomials, but $P_3(n)=n^2+1$ is not. The purpose of this note is to deduce, using results of…

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…

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection…

Two sets $\mathscr{A}$ and $\mathscr{B}$ are said to be cross-intersecting if $X\cap Y\neq\emptyset$ for all $X\in\mathscr{A}$ and $Y\in\mathscr{B}$. Given two cross-intersecting Sperner families (or antichains) $\mathscr{A}$ and…

A family $\mathcal{G}$ of sets is a weak copy of a poset $P$ if there is a bijection $f:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ implies $f(p)\subseteq f(q)$. If $f$ satisfies $p\leqslant q$ if and only if $f(p)\subseteq f(q)$,…

A nonempty set $A\subset\mathbb{N}$ is $\ell$-strong Schreier if $\min A\geqslant \ell|A|-\ell+1$. We define a set of positive integers to be sparse if either the set has at most two numbers or the differences between consecutive numbers in…

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE…

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…

We call $(a_1, \dots, a_n)$ an \emph{$r$-partial sequence} if exactly $r$ of its entries are positive integers and the rest are all zero. For ${\bf c} = (c_1, \dots, c_n)$ with $1 \leq c_1 \leq \dots \leq c_n$, let $S_{\bf c}^{(r)}$ be the…

A famous conjecture of Ryser states that any $r$-partite set system has transversal number at most $r-1$ times their matching number. This conjecture is only known to be true for $r\leq3$ in general, for $r\leq5$ if the set system is…

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R)^*$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is…

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is a subset…

Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq…

Let $G$ be a graph, $r \geq t$ integers, and $N \subseteq E(G)$. An $(r,t)$-threshold-coloring of $G$ with respect to $N$ is a mapping $c: V(G) \rightarrow \{0,\ldots,r-1\}$ such that $|c(u)-c(v)| \leq t$ for every $uv \in N$ and…

A conflict-free $k$-coloring of a graph $G=(V,E)$ assigns one of $k$ different colors to some of the vertices such that, for every vertex $v$, there is a color that is assigned to exactly one vertex among $v$ and $v$'s neighbors. Such…

Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $\sigma$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in \sigma} A_i \neq \emptyset$. Let…

A family of sets is said to be intersecting if every pair of sets in the family have non-empty intersection. In this paper, we initiate the study of intersecting non-uniform families of sets of one of two sizes containing given subfamilies.…

We prove an inverse of Furstenberg's correspondence principle stating that for all measure preserving systems $(X,\mu,T)$ and $A\subset X$ measurable there exists a set $E \subset \mathbb{N}$ such that \[ \mu\left( \bigcap_{i=1}^k…

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…

Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $\gamma >0$ (independent of…