  Envy-freeness is one of the most widely studied notions in fair division.
Since envy-free allocations do not always exist when items are indivisible,
several relaxations have been considered. Among them, possibly the most
compelling concept is envy-freeness up to any item (EFX). We study the
existence of EFX allocations for general valuations. The existence of EFX
allocations is a major open problem. For general valuations, it is known that
an EFX allocation always exists (i) when $n=2$ or (ii) when all agents have
identical valuations, where $n$ is the number of agents. it is also known that
an EFX allocation always exists when one can leave at most $n-1$ items
unallocated.
  We develop new techniques and extend some results of additive valuations to
general valuations on the existence of EFX allocations. We show that an EFX
allocation always exists (i) when all agents have one of two general valuations
or (ii) when the number of items is at most $n+3$. We also show that an EFX
allocation always exists when one can leave at most $n-2$ items unallocated. In
addition to the positive results, we construct an instance with $n=3$ in which
an existing approach does not work as it is.
