AP Free Sequences

In a recent [breakthrough](https://www.quantamagazine.org/surprise-computer-science-proof-stuns-mathematicians-20230321/), sequences which do not have Arithmetic Progressions (APs) of size 3, were studied. Let us try a simple problem related to that. Consider a sequence $A_1 \leq A_2 \leq \ldots \leq A_n$. Three elements from it, say $A_i, A_j, A_k$, where $i < j < k$, are said to form an AP, if