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First, suppose $G$ has a connected component $C\subseteq G$ with the same chromatic number as $G$. If $C\neq G$, we can take $M=C$. If $G=C$, let $M$ be the subgraph of $G$ obtained by removing all …

Yes, it is. The idea is that given an element of $\mathcal{M}$, you can detect whether the last step of it was $L$ or $U$, and then undo the steps one by one to recover a unique expression for it. F …

In any category, if $G$ and $H$ are objects such that there exist monic maps $G\to H$ and $H\to G$, then $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all $X$. There are plenty of pairs of non-isomor …

Note first that $L$ is naturally an endofunctor on the category of graphs and injective graph-homomorphisms that commutes with filtered colimits. Let $G$ be any graph such that there is an embedding …

A monotone map $f:[n]\to[m]$ is partition preserving if for all $i\in[n]$, $i\in t_+$ implies $f(i)\in t_+$ and $i\in t_-$ implies $f(i)\in t_-$. More simply, this means that the inverse image of eve …

No, it is not; here is a slightly simpler argument than the one indicated in my comment. As in the answer to the previous question, let $X=\Delta_3/\partial\Delta_3$. It suffices to show that the ca …