  We study unextendible maximally entangled basis in arbitrary bipartite
spaces. A systematic way of constructing a set of $d^{2}$ orthonormal maximally
entangled states in
$\mathbb{C}^{d}\bigotimes\mathbb{C}^{d'}(\frac{d'}{2}<d<d')$ is provided. The
complementary space of the set of these $d^{2}$ orthonormal maximally entangled
states contains no maximally entangled states that are orthogonal to all of
them. Furthermore, we investigate mutually unbiased bases in which all the
bases are unextendible maximally entangled ones. We present two unextendible
maximally entangled bases in $\mathbb{C}^{2}\bigotimes\mathbb{C}^{3}$ which are
mutually unbiased.
