Nonuniformity of downwards density in the $n$-computably enumerable Turing degrees

In 1993 R. Downey and M. Stob showed that downwards density of computably enumerable (further, c.e.) Turing degrees in the partial order of $2$-c.e. Turing degrees cannot be proved by a uniform construction. In this paper their result is generalized for any $n > 2$ and it is shown that there is no a uniform consruction for the downwards denstiy of $(n-1)$-c.e. degrees in the structure of $n$-c.e. degrees. Moreover, it is shown that there is no a uniform construction for the downwards denstiy in the structure of $n$-c.e. degrees.