All Questions

Let $G=(V,E)$ be a simple, undirected graph. Suppose that ${\cal S}$ is a collection of non-empty, connected, and pairwise disjoint subsets of $V$. Let $G({\cal S})$ be the graph with vertex set ${\...

Let $x,y\in \mathbb{Z}^\omega$ and let $x,y\in\mathbb{Z}^\omega$ form an edge if there is $i\in\omega$ such that $|x_i - y_i|=1$ and $ x_k = y_k$ for all $k\in \omega\setminus\{i\}$. $K_\omega$, the ...

Let $K$ be an infinite cardinal. Then, by the Robertson–Seymour theorem, the set of graphs with fewer than $K$ vertices and edges form a well-quasi-order. In terms of $K$, what is the maximal order ...

Let $G=(V,E)$ be an infinite simple, undirected graph with $\chi(G) \geq \aleph_0$. Is there a minor $M$ of $G$ such that $M\not\cong G$, and $\chi(M)=\chi(G)$ ?