  For a subset $A$ of an abelian group $G$, given its size $|A|$, its doubling
$\kappa=|A+A|/|A|$, and a parameter $s$ which is small compared to $|A|$, we
study the size of the largest sumset $A+A'$ that can be guaranteed for a subset
$A'$ of $A$ of size at most $s$. We show that a subset $A'\subseteq A$ of size
at most $s$ can be found so that $|A+A'| = \Omega(\min(\kappa^{1/3},s)|A|)$.
Thus a sumset significantly larger than the Cauchy--Davenport bound can be
guaranteed by a bounded size subset assuming that the doubling $\kappa$ is
large. Building up on the same ideas, we resolve a conjecture of Bollob\'as,
Leader and Tiba that for subsets $A,B$ of $\mathbb{Z}_p$ of size at most
$\alpha p$ for an appropriate constant $\alpha>0$, one only needs three
elements $b_1,b_2,b_3\in B$ to guarantee $|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$.
Allowing the use of larger subsets $A'$, we show that for sets $A$ of bounded
doubling, one only needs a subset $A'$ with $o(|A|)$ elements to guarantee that
$A+A'=A+A$. We also address another conjecture and a question raised by
Bollob\'as, Leader and Tiba on high-dimensional analogs and sets whose sumset
cannot be saturated by a bounded size subset.
