  The problem of fair division of indivisible goods has been receiving much
attention recently. The prominent metric of envy-freeness can always be
satisfied in the divisible goods setting (see for example \cite{BT95}), but
often cannot be satisfied in the indivisible goods setting. This has led to
many relaxations thereof being introduced. We study the existence of {\em
maximin share (MMS)} allocations, which is one such relaxation. Previous work
has shown that MMS allocations are guaranteed to exist for all instances with
$n$ players and $m$ goods if $m \leq n+4$. We extend this guarantee to the case
of $m = n+5$ and show that the same guarantee fails for $m = n+6$.
