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We say a set $A\subseteq\mathbb{Z}$ is arithmetical if there are integers $a>0,b\geq 0$ such that $A=\{ax+b:x\in\mathbb{Z}\}$. Is there $S\subseteq\mathbb{Z}$ such that $$S\cap A\neq\varnothing \neq ( …

For $\alpha,\beta\in \omega$ we set the absolute difference of $\alpha,\beta$ to be $$\lVert\alpha - \beta\rVert := |(\alpha\setminus\beta)\cup (\beta\setminus\alpha)|.$$ The absolute difference $\lVe …

A covering of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal …

A covering of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal …

Let $X$ be an infinite set, and let ${\cal E}$ be a collection of non-empty subsets of $X$. We say that ${\cal E}$ has property $\mathbf{B}$ if there is $B\subseteq X$ such that $B\cap E\neq \emptyset …

For every set $X$, let $[X]^2=\{\{x,y\}: x\neq y\in X\}$. Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. Note that, thanks to @Wojowu's comment below, the following holds. If …

For any set $X$ and cardinal $\kappa$, we denote by $\ [X]^\kappa :=\ \binom X\kappa\ $ the collection of subsets of $X$ having cardinality $\kappa$. If $X$ is a set, we call a set system $E\subseteq …

Let $X$ be an infinite set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. For $f\in\text{End}(X)$ let $$\text{fix}(f) = \{x\in X: x = f(x)\},$$ and $$\text{Com}(f) = \{g\in\text{End}( …

Let $X$ be an infinite set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. For $f\in\text{End}(X)$ let $$\text{Com}(f) = \{g\in\text{End}(X): g\circ f = f \circ g\}.$$ Is there $f\in \ …

Suppose that $L\subseteq {\cal P}(\omega)$ has the following properties: $\omega \notin L$, and for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then $|e_1\cap e_2|\leq 1$; if $m,n\in \omega …

Let ${\cal U}$ be a non-principal ultrafilter on $\omega$. If $\kappa>0$ is a cardinal, we say that a function $c:\omega \to \kappa$ is a coloring for ${\cal U}$ if for all $U\in{\cal U}$ the restrict …

For $a,b \in \omega$ with $a > 0$, let $f_{a,b}: \omega\to\omega$ be defined by $n \mapsto an+b$. What is an example of an infinite binary string $s:\omega\to\{0,1\}$ with the following property? Whe …

If $H=(V,E)$ is a hypergraph and $\kappa$ is a cardinal,we say a map $c:V\to\kappa$ is a coloring if the restriction $c\restriction_e$ of $c$ to $e$ is non-constant whenever $e\in E$ and $|e|>1$. The …

We say that two disjoint, non-empty subsets $S, T$ of an infinite cardinal $\kappa$ are neighboring if there is $\alpha\in \kappa$ such that $$S\cap\{\alpha,\alpha+1\} \neq \varnothing \neq T\cap\{\al …

A hypergraph $H=(V,E)$ is said to be complete and linear if whenever $e_1\neq e_2\in E$ then $|e_1\cap e_2|=1$, and for $v,w\in V$ there is $e\in E$ such that $\{v,w\}\subseteq e$. Assuming that $V$ …