The dimension of an ordered set 
is the least natural number
t
such that there exist linear orders 
on P extending
with 
. We write 
or simply dim(P)
for the dimension of P. The following fact is useful: Call an ordered pair
(x,y) of elements of P critical iff 
and for all 
,
u<x implies u<y and v>y implies v>x. Then dim(P) is the least natural
number t such that there exist linear orders on P
extending with y<x in 
for at least one i for each critical
pair (x,y) of P. Chapter 1 of [48] is a general refernce for all
these concepts.
The concept of dimension and Boolean layer cakes were born as twins: In
[19], Dushnik and Miller defined dimension and proved that P(1,n-1;n)
has dimension n whenever 
. In fact, P(1,n-1;n) is known today as
the
standard example of an n-dimensional order. Moreover, P(1,n-1;n) is
n-irreducible for , that is, dim(X)<n for any proper subset
under the induced order.
To determine the dimension of a Boolean layer cake is a challenging problem.
All known results deal with 2-layer cakes P(i,j;n), 
.
However, many other cases reduce easily to this setting:
Proof:
Certainly 
as the latter order
is a suborder of the former. On the other hand, every critical pair of
is in P(k,k+s;n), giving the reverse inequality:
Indeed, assume 
and k<|Y|<k+s. Since k+s<n, we
find two distinct points 
. Now if 
and
, then either 
or 
which shows that Y can't be the second component of a critical pair. The dual
argument shows it can't be the first, either.
To ease notation, we write dim(i,j;n) instead of dim(P(i,j;n)) and assume
tacitly that 0<i<j<n (it is easy to see that dim(0,n;n)=1 and
dim(0,j;n)=dim(i,n;n)=2 for all 0<i,j<n). The critical pairs of P(i,j;n)
are then all pairs (X,Y) with |X|=i, |Y|=j and 
. Until
recently, only the case i=1 received
serious attention, and it seems that little of what is known in this case
carries over to the 
setting. Accordingly, we first review the main
results for i=1.
