About I(q) definition- can a variable appear more than once at a CI (for example, x1_x2,x3|x3 or x1,x3_x2,x3|x1,x3)? or its just of the form x1_x2|x3? What about the regular pairwise independent (x1_x2|<nothing>)?

In other words, what is the maximal size of I(q) over n variables?

Moreover, are we expected to give an elegant way to prove such independents without enumerate over the options-table?

If so, may I get a clue?