The results for i>1 we are going to sample are all very recent. In fact, they are all contained in (or accessible through) papers coauthored by Brightwell, Füredi, Hurlbert, Kierstead, Kostochka, Talysheva and Trotter; these papers all appeared in Volume 11 of ORDER in 1994. See [5], [21] and [32].
We start with a formal analogue to Dushnik's and Miller's original result on
P(1,n-1;n). In [32], it is shown that dim(2,n-2;n)=n-2 and that,
moreover, the order p(2,n-2;n) is (n-1)-irreducible for all 
. The
analogy even extends to Dushnik's analysis of dim(1,j;n) for j large
compared to n: For , [32] provides conditions on
dim(2,j;n) which (almost) give the exact value of dim(2,j;n) for j
exceeding some critical value 
(with 
). `Almost' here
means that for certain exceptional values of j, 
, the estimate
obtained is only accurate to within 1.
The same paper ([32]) credits to Kostochka and Talysheva a proof of
dim(3,n-3;n)=n-2, valid for 
.
Extending these results, Füredi showed in [21] that generally one has
for all n>2i. The proof relies on the
Lovász-Kneser graph theorem. In fact, the main theorem of [21]
gives the exact value of dim(i,n-i;n) for all large n: For 
and
, we have dim(i,n-i;n)=n-2. The proof is based on a new variant of
the Erdös-Ko-Rado theorem.
The final group of results we are going to report deal with dim(i,i+k;n) for k relatively small. All are from [5]:
The following result provides an upper bound on dim(i,i+k;n) in terms of
values of dim(1,j;n): If 
and 
, then 
. Since 
(see
[23]), this establishes 
provided .
For k=1 and 
, a better estimate is possible: 
. The
authors credit Kostochka with establishing 
. For the most interesting case, that of the middle two layers of a Boolean
lattice with an odd number of atoms, the following upper and lower bounds are
given: Let n=2i+1, then 
.