  Properties of stable matchings in the popular random-matching-market model
have been studied for over 50 years. In a random matching market, each agent
has complete preferences drawn uniformly and independently at random. Wilson
(1972), Knuth (1976) and Pittel (1989) proved that in balanced random matching
markets, the proposers are matched to their $\ln n$th choice on average. In
this paper, we consider markets where agents have partial (truncated)
preferences, that is, the proposers only rank their top $d$ partners. Despite
the long history of the problem, the following fundamental question remained
unanswered: \emph{what is the smallest value of $d$ that results in a perfect
stable matching with high probability?} In this paper, we answer this question
exactly -- we prove that a degree of $\ln^2 n$ is necessary and sufficient.
That is, we show that if $d < (1-\epsilon) \ln^2 n$ then no stable matching is
perfect and if $d > (1+ \epsilon) \ln^2 n$, then every stable matching is
perfect with high probability. This settles a recent conjecture by Kanoria, Min
and Qian (2021).
  We generalize this threshold for unbalanced markets: we consider a matching
market with $n$ agents on the shorter side and $n(\alpha+1)$ agents on the
longer side. We show that for markets with $\alpha =o(1)$, the sharp threshold
characterizing the existence of perfect stable matching occurs when $d$ is $\ln
n \cdot \ln \left(\frac{1 + \alpha}{\alpha + (1/n(\alpha+1))} \right)$.
  Finally, we extend the line of work studying the effect of imbalance on the
expected rank of the proposers (termed the ``stark effect of competition''). We
establish the regime in unbalanced markets that forces this stark effect to
take shape in markets with partial preferences.
