MathOverflow

It is well known that $\chi(\mathbb{R}^2)\in\{5,6,7\}$. Fix $a\in\{5,6,7\}$. To prove $\chi(\mathbb{R}^2)\leq a$, according to the De Bruijn–Erdős theorem it is enough to prove that $\chi(G)\leq a$ for every finite subgraph of $\mathbb{R}^2$. Given a...

Let $f$ be an irreducible, non-linear polynomial in $\mathbb{Z[x]}$, and for every $d>1$ there exists a $n$ with $d\nmid{f(n)}$. In other words, It satisfies the Bunyakovsky condition. Let $p$ be a prime number such that $p\nmid{2\Delta_{f}}$. Give...

Let $k$ be a positive integer. Let $n$ and $m$ be positive integers with $m$ roughly $\log_2 n$, let $V$ be an $n$-dimensional vector space over $\mathbb{F}_2$, let $W$ be an $m$-dimensional vector space over $\mathbb{F}_2$, where $\mathbb{F}_2$ is the...

Assign exactly one $0$ and one $1$ to the two ends of each edge of a regular simple undirected bipartite graph of $e$ edges, uniform degree $d$ and number of vertices $2v$. The assignment of the pair $\{0,1\}$ to an edge can be considered as orienting...