  The existence of EFX allocations is a fundamental open problem in discrete
fair division. Given a set of agents and indivisible goods, the goal is to
determine the existence of an allocation where no agent envies another
following the removal of any single good from the other agent's bundle. Since
the general problem has been illusive, progress is made on two fronts: $(i)$
proving existence when the number of agents is small, $(ii)$ proving existence
of relaxations of EFX. In this paper, we improve results on both fronts (and
simplify in one of the cases).
  We prove the existence of EFX allocations with three agents, restricting only
one agent to have an MMS-feasible valuation function (a strict generalization
of nice-cancelable valuation functions introduced by Berger et al. which
subsumes additive, budget-additive and unit demand valuation functions). The
other agents may have any monotone valuation functions. Our proof technique is
significantly simpler and shorter than the proof by Chaudhury et al. on
existence of EFX allocations when there are three agents with additive
valuation functions and therefore more accessible.
  Secondly, we consider relaxations of EFX allocations, namely, approximate-EFX
allocations and EFX allocations with few unallocated goods (charity). Chaudhury
et al. showed the existence of $(1-\epsilon)$-EFX allocation with
$O((n/\epsilon)^{\frac{4}{5}})$ charity by establishing a connection to a
problem in extremal combinatorics. We improve their result and prove the
existence of $(1-\epsilon)$-EFX allocations with $\tilde{O}((n/
\epsilon)^{\frac{1}{2}})$ charity. In fact, some of our techniques can be used
to prove improved upper-bounds on a problem in zero-sum combinatorics
introduced by Alon and Krivelevich.
