On 2-Selmer ranks of quadratic twists of elliptic curves

We study the 2-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve E over an arbitrary number field K, if the set AE of 2-Selmer ranks of quadratic twists of E contains an integer c, it contains all integers larger than c and having the same parity as c. We also find suficient conditions on AE such that AE is equal to Z_≥tE for some number tE. When all points in E[2] are rational, we give an upper bound for tE.