  We study the problem of fairly allocating indivisible goods and focus on the
classic fairness notion of proportionality. The indivisibility of the goods is
long known to pose highly non-trivial obstacles to achieving fairness, and a
very vibrant line of research has aimed to circumvent them using appropriate
notions of approximate fairness. Recent work has established that even
approximate versions of proportionality (PROPx) may be impossible to achieve
even for small instances, while the best known achievable approximations
(PROP1) are much weaker. We introduce the notion of proportionality up to the
maximin item (PROPm) and show how to reach an allocation satisfying this notion
for any instance involving up to five agents with additive valuations. PROPm
provides a well-motivated middle-ground between PROP1 and PROPx, while also
capturing some elements of the well-studied maximin share (MMS) benchmark:
another relaxation of proportionality that has attracted a lot of attention.
