Let $\{X_i\}_{i=1}^{12}$ be the subjects, and $(a,b,c,d,e,f,g)$ be the students. If (in example) $(a,b) \in X_1$, then one of $(a,j), (b,j)$ is in $X_1$, for j = c,d,e,f,g. Our strategy is as follows. Pick a subject containing (a,b) [there is atleast one]. From our lemma above, we guarantee 5 other unique pairs in that subject. Now pick a subject containing a pair we havent got to yet. We guarantee 4 other unique pairs, because only one can repeat (it is easy to show **) Continuing in this fashion, we guarantee 6+5+4+3+2+1 = 21 unique pairs, and we are done. ** a short proof with handwaving: if in example we used (a,b), then we have also used either (x,a) or (x,b). So when we use (i, j), we use either (y,i) or (y,j). So if (x,a) = (y,i) in example, we have a,b,i,j,x fixed, so that when y varies it can only match up with (x,a) at most once.

Clarification -- what do the ordered pairs represent? And why would "only one of $(a,j),(b,j)$" be in $X_1$ if $(a,b) \in X_1$?