  We introduce the notion of the unextendible maximally entangled basis (UMEB),
a set of orthonormal maximally entangled states in d \times d consisting of
fewer that d^2 vectors which have no additional maximally entangled vectors
orthogonal to all of them. We prove that UMEBs don't not exist for d=2 and give
an explicit constructions for a 6-member UMEB with d=3 and a 12-member UMEB
with d=4.
