Erdős Problem #1: sum-distinct sets

The formal theorem proves that no universal constant $C>0$ can satisfy $$N>C\,2^{|A|}$$ for every nonempty interval bound $N$ and every sum-distinct $A\subseteq\{1,\dots,N\}$. Equivalently, for every $\varepsilon>0$ there are arbitrarily large $n$ and sum-distinct $n$-element sets contained in $\{1,\dots,N\}$ with $$N\le\varepsilon 2^n.$$ The proof is ineffective: it establishes the existence of arbitrarily large such $n$ but gives no explicit bound for how large $n$ must be in terms of $\varepsilon$.

The formal theorem proves that no universal constant $C>0$ can satisfy $$N>C\,2^{|A|}$$ for every nonempty interval bound $N$ and every sum-distinct $A\subseteq\{1,\dots,N\}$. Equivalently, for every $\varepsilon>0$ there are arbitrarily large $n$ and sum-distinct $n$-element sets contained in $\{1,\dots,N\}$ with $$N\le\varepsilon 2^n.$$ The proof is ineffective: it establishes the existence of arbitrarily large such $n$ but gives no explicit bound for how large $n$ must be in terms of $\varepsilon$.