Problem #301: Let $f(N)$ be the size of the largest $A\subseteq...

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Progress

The example $A=(N/2,N]\cap \mathbb{N}$ shows that $f(N)\geq N/2$.

Wouter van Doorn has given an elementary argument that proves\[f(N)\leq (25/28+o(1))N.\]Indeed, consider the sets $S_a=\{2a,3a,4a,6a,12a\}\cap [1,N]$ as $a$ ranges over all integers of the form $8^b9^cd$ with $(d,6)=1$. All such $S_a$ are disjoint and, if $A$ has no solutions to the given equation, then $A$ must omit at least two elements of $S_a$ when $a\leq N/12$ and at least one element of $S_a$ when $N/12<a\leq N/6$, and an elementary calculation concludes the proof.

Stijn Cambie and Wouter van Doorn have noted that, if we allow solutions to this equation with non-distinct $b_i$, then the size of the maximal set is at most $N/2$. Indeed, this is the classical threshold for the existence of some distinct $a,b\in A$ such that $a\mid b$.

See also [302] and [327].