<s>[INST] List of integers without any arithmetic progression of n terms Let's consider a positive integer $n$ and the list of the $n^2$ integers from $1$ to $n^2$. What is the minimum number $f(n)$ of integers to be cancelled in this list so that it is impossible to form  any arithmetic progression of $n$ terms with the remaining integers?
 [/INST] Extending my comment to an answer for small $n$: a brute force program gives
$$
\begin{align}
f(1)&=1 &&\{1\}  \\ 
f(2)&=3 &&\{1,2,3\} \\
f(3)&=4 &&\{3,4,5,7\} \\
f(4)&=6 &&\{3,4,5,6,10,13\} \\
f(5)&=7 &&\{3, 7, 9, 10, 11, 16, 21\} \\
f(6)&=9 &&\{5, 8, 12, 14, 15, 16, 21, 26, 31\} \\
f(7)&=11 &&\{3, 9, 11, 12, 13, 14, 15, 22, 29, 36, 43\} \\
f(8)&=13 &&\{7, 10, 16, 18, 19, 20, 21, 22, 29, 36, 43, 50, 57\}
\end{align}
$$
</s>