Using notation established in the previous subsection, let
be the number
of maximal proper sublattices of 
and write 
for the ratio
. Our first objective is to prove
The following corollary is immediate, settling the question posed at the end of subsection 3.1 (see also [2]):
Proposition 3.3.1 is proven by counting the large maximal sublattices of
type 
of 
as provided by Corollary
3.2.10.
Suppose 
, k=2i>0 even. Denote by 
the number of
distinct partitions of a k-element set A into 2-element subsets.
Proof:
This is obviously true for k=4. Suppose A' has k'=2i+2 elements
and the formula holds for k=2i. For any given 2-element subset
, there are many partitions of the required type
including 
. The number of such subsets is 
, so we
obtain 
suitable
partitions
of A'. But each of these has been counted k'/2 times (= the number of
blocks in any such partition), so 
which simplifies to
as required.
Proof of Proposition 3.3.1:
Assume that 
, 
is odd. We count the number of large
maximal sublattices of type of 
: For each 
, there are
as many of these as there are partitions of 
into 2-element
subsets; hence their total number is
. We
conclude that 
. On the other hand, we have 
whenever 
. Hence
Now 
whenever . We infer that 
for . It is obvious that
is not bounded from above for any
fixed exponent k, completing the proof.
Next, we use Theorem 3.2.11 to shed some light on the so-called
sublattice spectrum question raised first by Birkhoff in the 1948 edition of
his ``Lattice Theory''. In the wording of [4, p. 19,], the question
reads as follows: ``Given n, what is the smallest integer 
such that
every lattice with order 
contains a sublattice of exactly n
elements?''
It is shown by Havas and Ward in [27] that indeed exists for
any (n>0) and that 
for n>1. Not much seems to
be known about the values of 
otherwise.
Theorem 3.2.11 provides the possible sizes of large sublattices of
for n>6: These are either of the form 
or ,
with respective sizes 
and 
. Now 
, so contains no sublattices of size s for
and for
. It follows that
must be bigger than 
.
