Given $n\in\mathbb{N}$, what is the minimum value of $m$ for which we can color $1,2,\dots,n$ in $m$ colors such that no three numbers of the same color form an arithmetic progression?
Let $f(n)$ denote the specified value. For small values of $n$, we have $f(1)=f(2)=1$, $f(n)=2$ for $3\leq n\leq 8$, and I believe $f(9)=3$. Might the answer be $\lfloor\log_3n\rfloor+1$?