I am aware of the related question "Minimal size of an open affine cover", but would like to ask more specifically:
Do you have some elementary (i.e. not using hard things like compactification and such) proof for one of the following (here "variety" is separated over alg. closed field):
(1) Let $X$ be a variety; Can you show that $X$ can be covered by $C \cdot dim(X) + D$ open affines, where $C,D$ are universal constants?
(2) Let $X$ be quasi-projective; Can you show (1) for it with $C=1,D=1$?
(3) Let $X$ be smooth quasi-projective, and char. = 0; Can you show (2) for it?
It is easy for a variety $X$ to find an open affine whose complement is of smaller dimension than $X$. But I don't see how given $Y$ closed in $X$, to find an affine open $U$ in $X$ such that $Y-U$ is of smaller dimension than $Y$.