Avoidable structures, II: finite distributive lattices and nicely ...

AVOIDABLE STRUCTURES, II 23
for the ordered sequence of all members of Z ∪{b ij ,c ij } in the interval I[b ij ,c ij ].
We define the system S ′ to be the same as S except that the intervals will be
all the undivided intervals I[b i ,c i ] of S, together with all the intervals I[fr,f j j r+1 ]
where 0 ≤ j < u and 0 ≤ r < m j −1.
This is a system for P (easy to show). The divided intervals in S ′ are all among
the intervals I[d 0 x,d 1 x] and I[e 0 x,d 1 x] and each such interval has primitive interior
(since the endpoints of the interval are successive cut-points of a larger interval and
the interval is divided).
We added the set Z of points to the universe of the underlying basic skeleton in
order to generate these intervals. Thus S ′ is a (5M)-system for P. •
Lemma 5.11. Let P be a primitive ordered set in P N , N ≥ 3. Let k ≥ 1, and let
S be an (M)-system for P in which all the divided intervals are critical intervals
of depth k in P. Then P admits an (18N ·M)-system S ′′ such that every divided
interval of S ′ is a critical interval in P of depth k+1. Moreover, if statements (i)
and (ii) of Theorem 5.7 are true of S, they are true of S ′′ as well.
Proof. First we produce the (5M)-system S ′ for P, from Lemma 5.10.
Let I[c,c ′ ] be any one of the divided intervals of S ′ ; say c,c ′ are successive cutpoints
of the critical interval I[b ij ,c ij ] of depth k in P. (See the proof of Lemma
5.10). By Lemma 4.10, I(c,c ′ ) is either a ladder L m , or else h ⋄ (I(c,c ′ )) ≤ 2N +2.
By Proposition 4.8, if I(c,c ′ ) is a ladder L m , then m ≤ N +2. Thus in both cases,
h ⋄ (I(c,c ′ )) ≤ 2N +2.
Now we apply Theorem 3.12 to the primitive ordered set I(c,c ′ ) to obtain an
aligned tower Q 0 ,...,Q m−1 for this set. Here m ≤ 2N +2.
Thus for every divided interval in S ′ we have an aligned tower of at most 2N +2
blocks.
To construct S ′′ , we take X ′′ to be the union of X ′ = X and all the elements
in all these intervals I(c,c ′ ), excluding the members of the chambers of the long
blocks in these various towers.
For the intervals in S ′′ we take the undivided intervals of S together with the
chambers of the long blocks in the chosen aligned towers for the divided intervals
of S.
The lower and upper ladders of S ′ , Y 0 and Y 1 , are unchanged.
The only possible divided intervals in S ′′ are some of the chambers of these long
blocks in the towers for divided intervals of S ′ . These intervals are, by definition,
critical intervals in P of depth k +1.
Finally, how large is the universe S ′′ of the underlying basic skeleton of S ′′ . We
have added points to X and points to define the new intervals. In each of the
divided intervals of S ′ , we have added at most 3·(2N +2) points altogether. The
number of those divided intervals is at most 2|X| ≤ 2M. (See the proof of Lemma
5.10). Thus S ′′ is an (M ′′ )-system where
M ′′ ≤ 5M +2|X|·3(2N +2) ≤ 5M +12M(N +1) = M(17+12N) ≤