Problem #37: We say that $A\subset \mathbb{N}$ is an essential component...

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Progress

The answer is no by Ruzsa [Ru87], who proved that if $A$ is an essential component then there exists some constant $c>0$ such that $\lvert A\cap \{1,\ldots,N\}\rvert \geq (\log N)^{1+c}$ for all large $N$.

Furthermore, Ruzsa proves that this is best possible, in that for any $c>0$ there exists an essential component $A$ for which $\lvert A\cap \{1,\ldots,N\}\rvert \leq (\log N)^{1+c}$ for all large $N$.

In [Ru99] Ruzsa states "The simplest set with a chance to be an essential component is the collection of numbers in the form $2^m3^n$ and Erdős often asked whether it is an essential component or not; I do not even have a plausible guess."