Suppose we have $n^2$ red points and $n(n-1)$ blue points in the plane in general position. Is it possible to find a subset $S$ of red points such that the convex hull of $S$ does not contain any blue points, where $|S|>n^{\epsilon}$? 



monochromatic subset
By coloring the Horton set with two colors, periodically mod 3 according to the $x$-coordinate, [Devillers et al.](https://doi.org/10.1016/S0925-7721(03)00013-0) obtained arbitrarily large bicolored point sets with no monochromatic empty convex $5$-gon (that is, monochromatic $5$-hole).


Using the fact that every set of 10 points in general position in the plane contains a $5$-hole, the bicolored Horton set has no monochromatic $10$-island (a set of 10 points of the same color such that their convex hull contains no other points of the Horton set). 


So for sufficiently large $n$ the answer is no.


See <http://www.eurogiga-compose.eu/posezo/horton_set> for some additional references about the Horton set.


**Edit:**
The Horton sets have size $2^m$, and the coloring by Devillers et al. has one third of the points red and two thirds blue. Take $m$ such that $2^m>3n^2$.
To get $n^2$ red and $n(n-1)$ blue points, we take a second copy of the same Horton set of size $2^m$, with one third of the points blue and two thirds red, and place the two copies side by side, so that their convex hulls are disjoint. Then we cut off an appropriate number of points from the left copy ($n^2-2n$ if $n$ is divisible by $3$) and the right copy ($n^2+n$ if $n$ is divisible by $3$), by two vertical lines (or, in addition, remove a constant number of other points from the right side of the left copy or from the left side of the right copy to solve some divisibility issues). This will keep the maximum size of a monochromatic island constant (at most 19).


