Notation "set over integer"

Question

I'm reading a paper about graph theory and can't guess what could be meant with the Notation "$I\in\binom{V(G)\times V(H)}{\ell}$", where $\ell\in\mathbb{N}$ and $V(G),V(H)$ sets (of vertices of the graphs $G,H$). Later the notation "$\binom{V(G)\times V(H)}{\leq 2t}$" occurs for which I have no clue what it means either. I don't think the context is very helpful, it's probably just a common notation that I just don't know and couldn't find anything about it on the internet.

Answer 1

The notation

$$I\in\binom{V(G)\times V(H)}{\ell}$$

probably means "$I$ is a subset of $V(G)\times V(H)$, with cardinality $\ell$" (or, if $\ell$ is a set, "with cardinality equal to the cardinality of $\ell$).

This is a generalization of the binomial coefficient notation $\binom{a}{b}$. Instead of meaning "the number of subsets of cardinality $b$ of a set with cardinality $a$," it means "the collection of all subsets of cardinality $b$ of the set $a$."

Likewise, $\binom{V(G)\times V(H)}{\leq 2t}$ would be the collection of all subsets of $V(G)\times V(H)$ with cardinality less than or equal to $2 t$.