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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 mostcompelling concept is envy-freeness up to any item (EFX). We study theexistence of EFX allocations for general valuations. The existence of EFXallocations is a major open problem. For general valuations, it is known thatan EFX allocation always exists (i) when $n=2$ or (ii) when all agents haveidentical valuations, where $n$ is the number of agents. it is also known thatan EFX allocation always exists when one can leave at most $n-1$ itemsunallocated.  We develop new techniques and extend some results of additive valuations togeneral valuations on the existence of EFX allocations. We show that an EFXallocation always exists (i) when all agents have one of two general valuationsor (ii) when the number of items is at most $n+3$. We also show that an EFXallocation always exists when one can leave at most $n-2$ items unallocated. Inaddition to the positive results, we construct an instance with $n=3$ in whichan existing approach does not work as it is.

Extension of Additive Valuations to General Valuations on the Existence of EFX