ag.algebraic geometry - Minimal size of an open affine cover

This is not a complete answer by any means, but here are the two most basic arguments. First of all, you have that every projective scheme that can be embedded in $mathbb{P}^n$ can be covered by $n+1$ open affines, namely the closed subschemes of the affines $U_i = lbrace z_i neq 0 rbrace cong mathbb{A}^n$.

For a lower bound, think Cech-cohomologically: if $X$ can be covered by $k$ affine opens, then $check{H}^l(X) = 0$ for every $l > k$. If $X$ is Noetherian separated, then Cech cohomology coincides with sheaf cohomology, which indicated that you need at least $maxlbrace l ;|; H^l(X) neq 0 rbrace$ open affines to cover it.