Consider all the integer points $\in [0,n]\times[0,n]$, I want to find the maximum subset $S$ of which such that there are at most $n^\varepsilon(0<\varepsilon<1)$ points in $S$ collinear.

So, does any one can answer what magnitude of the size of $S$ will be?(write as the function $f(\varepsilon)$ of $\varepsilon$)