In a sequence of right-(k-1)-neighborhood-monochromatic vertices, how do we specify the $(k-1)$-sets that determine the colors of the $k$-sets?

Question

This is a piece of combinatorial reasoning in a proof I'm reading. I understand what "right-(k-1)-neighborhood-monochromatic" means, but how does one go from there to the number $[l-1] \choose k-1$ as in the picture below?

Answer 1

$[\ell -1] \choose k-1 $ is a set, not a number.

More precisely, it is the set of $(k-1)$-subsets of the integers from $1$ to $\ell-1$. You can discard $\ell$ because you know it will never be among the $k-1$ smallest indices of a $k$-subset.