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A subset $S\subseteq\mathbb{N}$ is said to be sum-free if whenever $s,t\in S$, then $s+t\notin S$. For instance the set of odd numbers is sum-free and has (lower and upper) asymptotic density 1/2. Que …

Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be …

Let $\newcommand{\Nplus}{\mathbb{N}^+}\Nplus$ denote the set of positive integers. Is there a function $f:\Nplus\to\Nplus$ with the following property? For all $(a,b)\in \Nplus\times\Nplus$ there is …

We say that a family ${\cal A}\subseteq {\cal P}(\omega)$ is Ramsey if for every map $c:{\cal A}\to\{0,1\}$ there is an infinite set $X\subseteq \omega$ with the following properties: ${\cal A}\cap { …

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 $\newcommand{\o}{\omega}\o$ be the set of non-negative integers, and for any set $X$, let $\newcommand{\oo}{[\o]^{<\o}}X^{<\o}$ denote the collection of all finite subsets of $X$. What is an examp …

Let $\newcommand{\oo}{[\omega]^{<\omega}}\oo$ denote the collection of finite subsets of the set of non-negative integers $\newcommand{\o}{\omega}\o$. If $A,B$ are any sets, let $A \,\triangle \, B = …

Let $\omega$ denote the set of non-negative integers. For which integers $n>1$ is there a sequence $b_n: \omega\to\omega$ with the following property? Whenever $v\in\omega^n$ there is a unique $i_v\i …

Let $\newcommand{\o}{[\omega]^\omega}\o$ denote the collection of infinite subsets of the set of nonnegative integers $\omega$. We say ${\cal A}\subseteq \o$ is almost disjoint if $A\cap B$ is finite …

Let $H=(V,E)$ be a hypergraph. We say that $H$ is thin if for every $v\in V$ the set $E_v=\{e\in E:v\in e\}$ is finite. A subset $D\subseteq V$ is dominating if $\bigcup \{e\in E:e\cap D \neq \emptyse …

Let $G=(V,E)$ be a simple, undirected graph. If $S, T\subseteq V$ are disjoint sets, we say that $S,T$ are connected to each other if there are $s\in S, t\in T$ such that $\{s,t\}\in E$. We say a grap …

Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are almost orthogonal if there is a positive integer $C_0\in \omega$ such that for all $n\in\om …

Motivation. Sometimes in life, people seem to do what the majority of their friends are doing. Do we all become more similar over time? Do we split up into pockets of similarity? This post aims to pro …

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\ …

This is inspired by an older, as of yet unanswered question. If $X$ is a set and $A,B\subseteq X$, we let the Hamming distance of $A, B$ be defined as $d_H:=\text{card}\big((A\setminus B)\cup (B\setmi …