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

24 W. DZIOBIAK, J. JEŽEK, AND R. MCKENZIE
M(6N +12N) = 18N ·M .
Proof of Proposition 5.8 and Theorem 5.7. The proposition follows immediately
by induction, from Lemmas 5.9 and 5.11. Thus, by this proposition, where
k = w(P), we get a ((18N) k−1 )-system S k−1 for P where each divided interval
(if any) is a critical interval of depth k −1. By Lemma 4.5, each of these divided
intervals I[b,c] has width at most 1; i.e., the interval is a chain. Thus the procedure
followed in the proof of Lemma 5.10 now produces a (5 · (18N) k−1 )-system S for
P that is nice (has no divided intervals). By Lemma 3.4 (3), we have k ≤ 6N +5.
This proves Theorem 5.7.
•
6. Avoiding {J k : k ≥ N}, III: conclusion
Theorem 6.1. For each N ≥ 1 the class A N is well-quasi-orderd by embeddability.
Proof. We can assume that N ≥ 3. Assume that this theorem is false. Choose, by
Theorem 1.1, a minimal bad sequence 〈L n : n < ω〉 in A N . By removing finitely
many terms from the sequence, we can assume that every L n is + o -indecomposable.
Let P n = L ∂ n for n < ω. The P n are primitive. By Theorem 5.7, P n is nicely (M)-
structured by a system S n , M = 5(18N) 6N+4 . By successively cutting down to
subsequences, we can assume that:
• There is a basic skeleton S b = (X;b 0 ,...,b m−1 ,c 0 ,...,c m−1 ), such that
every S n is an expansion (X,Y 0 ,Y 1 ;b 0 ,...,b m−1 ;c 0 ,...,c m−1 ) of S b .
• We have P n = S n (Q n 0,...,Q n m−1) where Q n j are convex subsets of P n with
Q n j ∂ < Pn ∂ ∼ = L n .
• The characters (¯c n 0,¯c n 1) of S n satisfy (¯c i 0,¯c i 1) ≤ (¯c j 0 ,¯cj 1 ) whenever i < j < ω.
Since Q n r ∂ < L n , then the collection {Q n r ∂ : 0 ≤ r < m,0 ≤ n < ω} is well-quasiordered.
Thus by successive further cutting down to infinite subsequences, we can
assume that
• When i < j we have Q i r∂
≤ Q
j
r
∂
for all 0 ≤ r < m.
But now, by Theorem 5.6, P0 ∂ ≤ P1 ∂ , equivalently, L 0 ≤ L 1 . This is a contradiction,
and it concludes our proof of the theorem.
•
Theorem 6.2. The unavoidable members of 〈D,≤〉 are precisely the lattices isomorphic
to a proper sublattice of some J n .
Proof. Since {J n : n < ω} is an anti-chain, any unavoidable finite distributive
lattice must be isomorphic to a proper sublattice of some J n . But we immediately
see that if L is isomorphic to a proper sublattice of J n , then L < J n+m for all
m ≥ 0. Thus in this case, {K ∈ D : L ≰ K} is contained in A n−1 and, by Theorem
6.1, is well-quasi-ordered. This means that L is unavoidable. •