@PeterLeFanuLumsdaine Ha-ha, I wasn't serious. If you want a more serious comment though, I don't think a rule that is "more logical" or "more reasonable" is better for the same reason prescriptive grammar sucks. Linguistics deals with descriptive grammar for a reason.

@PeterLeFanuLumsdaine Oh, you want a phonemic orthography?. That battle is already lost when you chose English.

@WłodzimierzHolsztyński You can find basic facts in the paper I linked to in my post (or those given by Thomas Kalinowski as well, I think). Handbook of Combinatorial Designs is a good reference book for this sort of basic knowledge; coverings are treated in Section 11 of Chapter IV in the 2nd edition. As for the typos, yes, they are not correct, although the number of $3$-subsets in this case is not $\frac{v(v-1)}{3}$ but $\frac{\binom{v}{2}}{\binom{3}{2}} = \frac{v(v-1)}{6}$. I'll edit OP's post.

Caro and Yuster's proof might be incomplete (although the statement itself is probably correct). See my answer (and the paper I linked to) for more detail.