A general overview of this case is provided in section 7.2 of Trotter's book [48]. Dushnik was the first to consider the problem in general: In [18], he investigated dim(1,j;n) for values of j large in comparison to n and obtained the following estimates:
Let s and t be positive integers satisfying 
and all of 
, 
, 
. Then
Dushnik also showed in [18] that the preceding inequalities determine
the exact
value of dim(1,j;n) whenever 
and 
. The critical value
is specified by the following conditions: Let 
be the largest
positive integer u with 
and satisfying 
for all 
, and put 
(thus 
).
Dushnik [18] and Trotter [47] use these estimates to give the
exact values of dim(1,j;n) for several cases where 
and 
, the values then are 
. See [32], Corollary 2.2 and
Theorem 3.3. [47] also contains a table of all values dim(1,i;n) for
.
If j is kept fixed, the asymptotic behavior of dim(1,j;n) as a function of
n was determined by Spencer in [46] by probabilistic methods:
for fixed 1<j<n. For j=2, Füredi,
Hajnal,
Rödl and Trotter provide the following asymptotic formula in [22]:
.
The most recent entry is [33, Theorem 0.7,] where Kierstead provides upper
and lower bounds for dim(1,j;n) which differ by a factor of at most
for j large (compared to n) for some constant c, and by
a factor of at most 
for j small (compared to n).