OPEN
For any $n$, let $A(n)=\{0<n<\cdots\}$ be the infinite sequence with $a_0=0$ and $a_1=n$, and for $k\geq 1$ we define $a_{k+1}$ as the least integer such that there is no three-term arithmetic progression in $\{a_0,\ldots,a_{k+1}\}$.
Can the $a_k$ be explicitly determined? How fast do they grow?
It is easy to see that $A(1)$ is the set of integers which have no 2 in their base 3 expansion. Odlyzko and Stanley have found similar characterisations are known for $A(3^k)$ and $A(2\cdot 3^k)$ for any $k\geq 0$, see
[OdSt78], and conjectured in general that such a sequence always eventually either satisfies
\[a_k\asymp k^{\log_23}\]
or
\[a_k \asymp \frac{k^2}{\log k}.\]
There is no known sequence which satisfies the second growth rate, but Lindhurst
[Li90] gives data which suggests that $A(4)$ has such growth ($A(4)$ is given as
A005487 in the OEIS).
Moy [Mo11] has proved that, for all such sequences, for all $\epsilon>0$, $a_k\leq (\frac{1}{2}+\epsilon)k^2$ for all sufficiently large $k$.
In general, sequences which begin with some initial segment and thereafter are continued in a greedy fashion to avoid three-term arithmetic progressions are known as Stanley sequences.
