  Given $n$ men, $n$ women, and $n$ dogs, each man has an incomplete preference
list of women, each woman does an incomplete preference list of dogs, and each
dog does an incomplete preference list of men. We understand a family as a
triple consisting of one man, one woman, and one dog such that each of them
enters in the preference list of the corresponding agent. We do a matching as a
collection of nonintersecting families (some agents, possibly, remain single).
A matching is said to be nonstable, if one can find a man, a woman, and a dog
which do not live together currently but each of them would become "happier" if
they do. Otherwise the matching is said to be stable (a weakly stable matching
in 3-DSMI-CYC problem). We give an example of this problem for $n=3$ where no
stable matching exists. Moreover, we prove the absence of such an example for
$n<3$. Such an example was known earlier only for $n=6$ (Biro, McDermid, 2010).
The constructed examples also allows one to decrease (in two times) the
dimension of the recently constructed analogous example for complete preference
lists (Lam, Plaxton, 2019).
