The following is a variation (something like it) of a marriage problem that I found interesting but couldn't quite get my head around on finding the conditions for this to work.
Let $B = \{b_1, . . . , b_k\}$ be a set of men and $G = \{g_1, . . . , g_n\}$ be a set of women, and suppose that each men wants to marry up to four of the women whom he fancies and who fancy him back (i.e. polygamy up to four wives is allowed, and man $b_i$ wants $0 \le n_i \le 4$ wives).
Find necessary and sufficient conditions for this problem to be solvable.
Of course the condition that every set of k men must fancy at least k women is needed for this problem to be solvable (ie: the case where every men can marry only 1 women). I'm guessing we can extend this to $ik$ women where $2 \le i \le 4$ for cases where every men can marry 2 women, 3 women, 4 women. However these do not seem like sufficient conditions for this problem to be solvable.
Are there other conditions that I am missing that seems needed?