  Erd\H{o}s asked the following question: given $n$ points in the plane in
almost general position (no 4 collinear), how large a set can we guarantee to
find that is in general position (no 3 collinear)? F\"uredi constructed a set
of $n$ points in almost general position with no more than $o(n)$ points in
general position. Cardinal, T\'oth and Wood extended this result to
$\mathbb{R}^3$, finding sets of $n$ points with no 5 on a plane whose subsets
with no 4 points on a plane have size $o(n)$, and asked the question for higher
dimensions: for given $n$, is it still true that the largest subset in general
position we can guarantee to find has size $o(n)$? We answer their question for
all $d$ and derive improved bounds for certain dimensions.
