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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 $G = (V,E)$ be a graph. Every clique, that is, complete subgraph, is contained in a maximal clique with respect to $\subseteq$ (this is an easy consequence of Zorn's Lemma). Let $\newcommand{\MC}{ …

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 …

Motivation. Suppose we are given $6$ boxes, arranged in the following manner: $$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]$$ Two of these boxes contain a present, and the rem …

Let $f:\mathbb{N} \to \{0,1\}$ be the Ehrenfeucht-Mycielski sequence. The first few digits of the sequence are: $$010011010111000100001111\ldots$$ For any $k\in\mathbb{N}$ let $s(k) = \sum_{i=0}^k f(i …

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 …

Starting point. Consider the "$V$-graph" on the vertex set $\{1,2,3\}$ and let the edges be $\{1,2\}$ and $\{2,3\}$. This graph is clearly bipartite. It is a trivial observation that whenever we color …

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 …