Explanation of definition of ${[n] \choose k}$

Question

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 ${[n] \choose k}$?

JMoravitz · Accepted Answer · 2016-04-13T04:46:58.127

2

$[n]=\{1,2,3,\dots,n\}$

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 $\{1,2,\dots,n\}$

edited Apr 13 '16 at 04:46

answered Apr 13 '16 at 04:40

JMoravitz

Thanks so much! I have an additional question, the part that says pairwise vertex disjoint paths means that no pair of paths share a vertex? – urpi Apr 13 '16 at 06:51
@urpi: That’s correct. – Brian M. Scott Apr 14 '16 at 06:05

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