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Let $\omega$ denote the set of non-negative integers. For sets $A,B$, let $B^A$ denote the set of maps $f:A\to B$. For $f,g\in\{-1,1\}^\omega$ we say that $f,g$ are almost orthogonal if there is $C_0\ …

If $X$ is an infinite set and $f:X\to X$, do we have $$\big|\{g:X\to X: g\circ f =f \circ g\}\big|\geq |X| ?$$

Motivation. A while ago I attended a party and I only knew some, but not all, of the attendees. There were 2 kinds of drinks: beer and soda. I noticed that amongst my acquaintances, more than half dra …

Let $X$ be a non-empty set, and let ${\cal S}\subseteq {\cal P}(X)$ be family of non-empty subsets of $X$. We say that ${\cal S}$ is intersecting if any two members of ${\cal S}$ have non-empty inters …

This is a follow-up question to an older question. Let $X\neq \emptyset$ be a set. We say that ${\cal C}\subseteq {\cal P}(X)$ is a cover if $\bigcup {\cal C} = X$, and we call ${\cal C}$ linear if $| …

For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a …

For any set $X$, let $\newcommand{\N}{\mathbb{N}}[X]^2 = \big\{\{x,y\}:x\neq y \in X\big\}$ and set $[n]^2 = [\{0,\dotsc,n-1\}]^2$ for any positive integer $n$. For $A\subseteq [\N]^2$ we set $$\newco …

Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say ${\cal A}\subseteq [\omega]^\omega$ is almost disjoint if $A \cap B$ is finite whenever $A\neq B \in {\cal A}$. Zorn …

Let $G = (V,E)$ be a simple, undirected graph with $\chi(G)$ infinite. Given a cardinal $\kappa$ with $0 < \kappa < \chi(G)$, is there an induced subgraph $S$ of $G$ with $\chi(S) = \kappa$? What I tr …

Let $\mathbb{N}$ denote the set of non-negative integers. We say $A\subseteq \mathbb{N}$ is a finite arithmetic progression if there are $a, n, d\in\mathbb{N}$ with $d \geq 1$ and $n \geq 2$ such that …

We say that a family $\mathcal R\subseteq \mathcal P(\omega)$ is Ramsey if $\bigcup \mathcal R = \omega$, and for every map $f:\mathcal R \to \{0,1\}$ there is an infinite set $X\subseteq \omega$ suc …

Let $X$ be a set, and let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ …

Let $\kappa$ be an infinite cardinal. We say $E\subseteq {\cal P}(\kappa)$ is an infinite projective plane on $\kappa$ if $e_1\neq e_2\in E$ implies $|e_1\cap e_2| = 1$, and whenever $n\neq m\in \ka …

If $G$ is a group and $S\subseteq G$, let $\langle S \rangle$ be the intersection of all subgroups of $G$ containing $S$. Let $S_\omega$ denote the group of all bijections $f:\omega\to\omega$ with co …

Let $X$ be an infinite set, and let ${\cal A}\subseteq{\cal P}(X)$ be a family of non-empty sets. We say $S\subseteq X$ is a cover for ${\cal A}$ if $A\cap S \neq \emptyset$ for all $A\in{\cal A}$. S …