  Maximin share (MMS) allocations are a popular relaxation of envy-free
allocations that have received wide attention in the context of the fair
division of indivisible items. Although MMS allocations can fail to exist [1],
previous work has found conditions under which they exist. Specifically, MMS
allocations exist whenever $m \leq n+5$ in the context of goods allocation, and
this bound is tight in the sense that MMS allocations can fail to exist when $m
= n+6$ [2]. Unfortunately, the technique used to establish this result does not
generalize readily to the chores and mixed manna settings. This paper
generalizes this result to the chores setting and provides a partial solution
for the mixed manna setting. Our results depend on the presence of certain
types of agents. Specifically, an agent $i$ is a goods agent (resp. chores
agent) if every item is a good (resp. chore) to $i$, and a non-negative mixed
agent if $i$ is neither a goods nor a chores agent and the MMS guarantee of $i$
is non-negative. In this paper, we prove that an MMS allocation exists if $m
\leq n+5$ and there exists a goods agent, a non-negative mixed agent, or only
chores agents.
  [1] David Kurokawa, Ariel D Procaccia, and Junxing Wang. When can the maximin
share guarantee be guaranteed? In Thirtieth AAAI Conference on Artificial
Intelligence, 2016.
  [2] Uriel Feige, Ariel Sapir, and Laliv Tauber. A tight negative example for
mms fair allocations. In International Conference on Web and Internet
Economics, pages 355-372. Springer, 2021.
