Question about notation, subsets of a graph and intersection of vertices

Question

I have the following description of a graph:

Let $G$ be a graph such that all of its vertices are subsets with two elements of $\{1,2,...,n\} (n\ge 2)$ where two sets $A,B$ are adjacent iff $A\cap B = \emptyset$.

I don't understand to express this graph, what does it mean with graphs $A\cap B = \emptyset$? how can an intersection not be empty?

For example:

$A=\{a,b\}, B=\{b,c\}: $ a___b___c is $A\cap B = \emptyset$?

Then how can there be $A\cap B \neq \emptyset$?

Note I study simple graphs so no double edges or edges that go into where it came from.

Answer 1

It may help to rewrite it.

Let us say a 2-subset is a subset with two elements, and let $n\geq 2$.

I would rewrite the statement in the following way.

Let $G$ be a graph such that all of its vertices are 2-subsets of $\{1,2,\ldots,n\}$, where two vertices $A$ and $B$ of $G$ are adjacent if and only if they are disjoint subsets of $\{1,2,\ldots,n\}$.

So vertices represent subsets, and two are connected if they do not intersect.

Hope this helps.

Answer 2

right, so if we're looking at $n=3$. Then our vertices must take the form of $(a,b)$ where $a \in {1,2,3}$ and $b \in {1,2,3}$. You can easily verify that any such vertices must not have a non-empty intersection as you have noted. However, if we bump up to $n = 4$. We can generate some vertices where there subsets have a non-empty intersection. For example, let $v_1 = \{1,4\}$ and $v_2 = \{2,3\}$. Then, since $\{1,4\} \cap \{2,3\} = \emptyset$. Thus, $v_1v_2 \in E(G)$. So there are examples where the intersection is empty.