I am reading a definition in a paper, but am not sure of how to interpret the following definition:
If $K \in {[n] \choose k}$, then let $\operatorname{Path}(K)$ denote the set
$$\{S: S \text{ is a set of pairwise vertex disjoint paths from} [k]\setminus(K \cap [k]) \text{ to }K\setminus([k] \cap K)\}\;.$$
Could someone explain me what the $[k]$ mean and how to interpret the ${[n] \choose k}$?
$[n]=\{1,2,3,\dots,n\}$
When $A$ is a set, $\binom{A}{r} = \{B~:~B\subseteq A,~|B|=r\}$
So, in this case, $\binom{[n]}{k}$ is the set of $k$-element subsets of $\{1,2,\dots,n\}$