In [Er73] and [Er77c] Erdős mentions an unpublished proof of Haight that \[\lim \frac{\lvert A\cap [1,x]\rvert}{x}=0\] holds if the elements of $A$ are independent over $\mathbb{Q}$.
Over the years Erdős asked for various different quantitative estimates, for example \[\liminf \frac{\lvert A\cap [1,x]\rvert}{x}=0\] or even (motivated by Behrend's bound) \[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}\ll \frac{\log x}{\sqrt{\log\log x}}.\] In [Er97c] he offers \$500 for resolving the questions in the main problem statement above.
This was partially resolved by Koukoulopoulos, Lamzouri, and Lichtman [KLL25], who proved that we must have \[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n).\]
See also [858].
This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
An example of such a set $A$ is the set of all integers in $[N^{1/2},N]$ divisible by some prime $>N^{1/2}$.
See also [143].
The game ends when no legal move is possible. One player wants the game to last as long as possible, the other wants the game to end quickly. How long can the game be guaranteed to last for?
At least $\epsilon n$ moves? (For $\epsilon>0$ and $n$ sufficiently large.) At least $(1-\epsilon)\frac{n}{2}$ moves?
This type of game is known as a saturation game.
In [Er98] Erdős reports (but gives no reference) that Sándor has proved that $\lvert A\rvert=(1-o(1))\log_2 n$ is achievable, taking $A=\{ 2^i+m2^m : 0\leq i<m\}$ and $n=2^{m-1}+m2^m$.
Erdős, Lev, Rauzy, Sándor, and Sárközy [ELRSS99] proved that \[\lvert A\rvert > \log_2 n -1\] is achievable, taking $A=\{2^m-2^{m-1},2^m-2^{m-2},\ldots,2^m-1\}$. This property also implies that $\sum_{a\in S}a$ are distinct for distinct subsets $S$, whence [1] implies \[\lvert A\rvert \leq \log_2 n+\tfrac{1}{2}\log_2\log n+O(1),\] and likely $\lvert A\rvert\leq \log_2n+O(1)$.
See also [13].
In particular, is this always possible if there are no non-trivial solutions to $(b_i,b_j)=b_k$?
One can ask a similar question for sequences of real numbers, as in [143].