Theorem 16
(Kruskal-Katona)
.
Assuming that:
A
⊂
X
(
r
)
,
1
≤
r
≤
n
C
is the initial segment of
colexicographic
on
X
(
r
)
with
|
C
|
=
|
A
|
Then
|
∂
C
|
≤
|
∂
A
|
. In particular: if
|
A
|
=
k
r
, then
|
∂
A
|
≥
k
r
−
1
.