A problem about how many evening we can meet all friends There are 30 friends and everyone lives in their own apartment. Everybody wants to
visit everybody else's apartment. Every evening a person can either stay home for the
whole evening, or visit as many friends as she likes. (Of course, if she visits somebody
who is not at home, we don't count it as a visit.) Prove that for each person to visit all
the others:
(1) 4 evenings are not enough;
(2) 5 evenings are also not enough;
(3) but 10 evenings is enough;
(4) and even 7 evenings is enough.
If the "visit" is a single direction, I can say in five days, each person can visit all other persons. However the problem seems to be two direction? how can we do?

I have worked out the problem . Actually it is very simple. We can let S be the days we consider. S_i be the set of visit days for each i. The problems say that S_i and S_j can not be the subset of the other. For 7 evenings, since 7 choose 3 equal 35, hence we can construct 30 different 3-tuple set.
