Fix $k\in\mathbb{N}.$ Definition: A set $A\subset\mathbb{N}$ (with $\vert A \vert = \infty$) is maximal of length $k$ if it contains at least one arithmetic progression of length $k,$ and, if $x\not\in A,$ then $A\cup\{x\}$ will contain an arithmetic progression of length $>k;$ equivalently, $A$ will no longer only contain A.P.'s of length $\leq k.$
An example of a Set $A$ which is maximal of length $2$ is the following "Greedy (algorithm)" Salem-Spencer set: $\quad$ A003278 .
My question is this:
Does there exist $A\subset\mathbb{N}$ that is maximal of length $k,$ such that if $x\not\in A,$ then $A\cup\{x\}$ will contain an arithmetic progression of length $\geq k+2 ?$
[This would mean that there is no $x\in A$ which (only) brings the length of the greatest A.P. of $A$ up to $k+1.$]
I'm guessing that a general example can be constructed, but I can't quite think how to do so.