  We show that for every positive integer R there exist monomial ideals
generated in degree two, with linear syzygies, and regularity of the quotient
equal to R. Such examples can not be found among Gorenstein ideals since the
regularity of their quotients is at most four. We also show that for most
monomial ideals generated in degree two and with linear syzygies the regularity
grows at most doubly logarithmically in the number of variables.
