Search Results

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 $G=(V,E)$ be a finite, simple, undirected graph. The Hadwiger number $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. Hadwiger's celebrated conjecture states that $\chi …

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 …

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 …

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 …