Yes, if the universe is:
-flat (zero spatial curvature)
-has finite mass energy (since we know it is uniform this also means it is bounded. If you drop the bounded es because you don't want to admit uniformity or otherwise, i.e., if it is unbounded, then the answer is clearly no)
-and is simply connected (has what is called a trivial topology)
The it does have to have an edge.
See the zero curvature and other sections of the wiki article on the shape of the universe, it's fairly complete, at https://en.m.wikipedia.org/wiki/Shape_of_the_universe
The simply connected condition is critical also. If you allow other topologies then both the torus and the Klein bottle topologies are bounded, flat and have no edges.