Erdős Problem #296 — Disjoint Unit-Fraction Decompositions

For a finite set $A\subseteq\mathbb N$, let

$$R(A)=\sum_{n\in A}\frac1n.$$

For natural numbers $N$ and $k$, say that there are $k$ disjoint unit decompositions in $\{1,\ldots,N\}$ if there exist pairwise disjoint finite subsets $A_1,\ldots,A_k\subseteq\{1,\ldots,N\}$ such that $R(A_i)=1$ for every $i$. Then both of the following hold:

1. For all natural numbers $N,k$, if there are $k$ such disjoint decompositions, then

$$k\leq\sum_{n=1}^{N}\frac1n.$$

2. For every real number $\varepsilon$ with $0<\varepsilon<1$, for all sufficiently large $N$ there are

$$\left\lfloor(1-\varepsilon)\log N\right\rfloor$$

pairwise disjoint subsets of $\{1,\ldots,N\}$, each having reciprocal sum $1$.