On further questions regarding unit fractions

We prove that a subset $A\subseteq [1,N]$ with $$\sum _{n\in A}\frac{1}{n}\ge (\log N{)}^{4\mathrm{/}5+o(1)}$$ contains a subset $B$ such that $$\sum _{n\in B}\frac{1}{n}=1.$$ Our techniques refine those of Croot and of Bloom. Using our refinements, we additionally consider a number of questions regarding unit fractions due to Erdős and Graham.