Table Of ContentGames on finitely generated structures
Adam Krawczyk Wies(cid:32)law Kubi´s∗
Insititute of Mathematics Institute of Mathematics
University of Warsaw, Poland Czech Academy of Sciences
Prague, Czechia
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——
1
0 Institute of Mathematics
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Cardinal Stefan Wyszyn´ski University
b Warsaw, Poland
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F
6 February 7, 2017 1x(cid:157)
]
O
L
h. Abstract
t
a We study the abstract Banach-Mazur game played with finitely generated struc-
m
tures instead of open sets. We characterize the existence of winning strategies aiming
[ at a single countably generated structure. We also introduce the concept of weak
Fra¨ıss´e classes, generalizing the classical Fra¨ıss´e theory, revealing its relations to our
2
Banach-Mazur game.
v
6
5 MSC (2010): 03C07, 03C50.
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5
Keywords: Banach-Mazur game, weak amalgamation, Fra¨ıss´e class.
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.
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1 Introduction
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1
:
v WeconsiderthefollowinginfinitegamefortwoplayersEve andOdd. Namely,Evestartsby
i
X choosing a small (typically: finitely generated) structure A0. Odd responds by choosing a
biggersmallstructureA ⊇ A . EverespondsbychoosingasmallstructureA containing
r 1 0 2
a
A . Andsoon;therulesforbothplayersremainunchanged. Specifically,wefixacountable
1
first-order structure G and denote, as usual, by Age(G) the class of all finitely generated
structures embeddable into G. The game BM(G) is played with structures from Age(G).
(cid:83)
We say that Odd wins if the union A is isomorphic to G; otherwise Eve wins.
n∈ω n
This is in fact an abstract version of the well-known Banach-Mazur game [7], in which
open sets are replaced by abstract objects. In [2] it was shown that Odd has a winning
strategy in BM(G) whenever Age(G) is a Fra¨ıss´e class (equivalently: G is homogeneous
with respect to its finitely generated substructures). The paper [2] contains also examples
of non-homogeneous graphs G for which Odd still has a winning strategy. One needs to
∗Research of the second author supported by GACˇR grant No. 17-27844S.
1
admit that our Banach-Mazur game is a special case of infinite games considered in model
theory. For more information, see the monograph of Hodges [4].
Our goal is to present a non-trivial characterization when Odd has a winning strategy
in BM(G), where G is a countable first-order structure. We also develop the theory of
limits of weak Fra¨ıss´e classes, where the amalgamation property is replaced by a weaker
conditionintroducedbyIvanov[5]. WeshowthatOddhasawinningstrategyinBM((G))
if and only if Age(G) is a weak Fra¨ıss´e class and G is its limit.
2 The setup
Throughout this note F will always denote a class, closed under isomorphisms, consisting
of countable finitely generated structures of a fixed first-order language. We will denote
by σF the class of all structures isomorphic to (cid:83) X , where {X } is an increasing
n∈ω n n n∈ω
chain of structures from F.
The relation X (cid:54) Y will mean, as usual, that X is a substructure of Y. We define the
hereditary closure of F by
F↓ = {X: (∃ Y ∈ F) X (cid:54) Y and X is finitely generated }.
Note that σ(F↓ ) may be strictly larger than σF (see Example 5.4 below). Recall that
F is hereditary if F↓ = F. Recall that F has the joint embedding property (JEP) if for
every X,Y ∈ F there is Z ∈ F such that X (cid:54) Z and Y (cid:54) Z.
We will consider the game BM(F,G) described in the Introduction, where both players
are allowed to play with structures from F and G ∈ σF. The following facts are rather
straightforward; for the Reader’s convenience we provide the proofs.
Proposition 2.1. Suppose Odd has a winning strategy in BM(F,G) for some G ∈ σF.
Then F has the JEP and contains countably many isomorphic types.
Proof. EvecanstartwithanarbitraryX ∈ F. AsOddhasawinningstrategy, wededuce
that G contains copies of all structures in F, while on the other hand G can contain only
countably many finitely generated substructures. The JEP also follows, because given
X,Y (cid:54) G = (cid:83) G with each G ∈ F, we can find m such that X,Y (cid:54) G , as X and
n∈ω n n m
Y are finitely generated.
Proposition 2.2. Let G ,G be such that Odd has winning strategies both in BM(F,G )
0 1 0
and BM(F,G ). Then G is isomorphic to G .
1 0 1
Proof. Let Σ be Odd’s winning strategy in BM(F,G ), where i = 0,1. Let us play the
i i
game assuming that Odd is using strategy Σ . Eve starts with some randomly chosen
0
A ∈ F and her choice A is computed according to strategy Σ . From that point on,
−1 0 1
Eve is using strategy Σ applied to sequences of the form
1
A (cid:54) A (cid:54) ... (cid:54) A ,
−1 0 n−1
(cid:83)
where n is even. In this situation both players win, showing that A = A is
∞ n∈ω n
isomorphic to both G and G .
0 1
2
In the sequel we shall frequently use the following trivial fact.
Lemma 2.3. Let A,B ∈ F, let h: A → B be an isomorphism, and let B(cid:48) ∈ F be such
that B(cid:48) (cid:62) B. Then there exist A(cid:48) (cid:62) A and an isomorphism h(cid:48): A(cid:48) → B(cid:48) extending h.
3 The weak extension property
As before, F is a fixed class of countable finitely generated structures, and σF is the
class of all unions of countable chains of structures from F. We shall say that G ∈ σF is
F-universal if every structure from F embeds into G.
Proposition 3.1. Let F be as above and let G ∈ σF. The following properties are
equivalent:
(a) For every A (cid:54) G, A ∈ F, there is B ∈ F such that A (cid:54) B (cid:54) G and for every
X ∈ F with B (cid:54) X there exists an embedding f: X → G satisfying f (cid:22) A = id .
A
(b) For every A (cid:54) G, A ∈ F, there are an isomorphism h: A(cid:48) → A and B (cid:62) A(cid:48),
B ∈ F, such that for every X ∈ F with B (cid:54) X there exists an embedding f: X → G
extending h.
(c) For every A ∈ F, for every embedding e: A → G there is B (cid:62) A, B ∈ F, such that
for every X ∈ F with B (cid:54) X there exists an embedding f: X → G extending e.
Proof. (a) =⇒ (b) Take h := id .
A
(b) =⇒ (c) Note that the assertion of (b) holds when the isomorphism h is replaced by
h ◦ g, where g: A(cid:48)(cid:48) → A(cid:48) is an arbitrary isomorphism. This follows immediately from
Lemma 2.3 applied to g−1. Thus, one can take g to be any automorphism of A(cid:48), therefore
h◦g can be an arbitrary embedding of A(cid:48) into G whose image is A, which shows (c).
(c) =⇒ (a) Take e := id and apply (c) obtaining a suitable B (cid:62) A. In particular, there
A
is an embedding g: B → G that is identity on A. Then g[B] ∈ F and A (cid:54) g[B] (cid:54) G. Fix
X ∈ F withg[B] (cid:54) X. ByLemma2.3,thereareX(cid:48) (cid:62) B andanisomorphismg(cid:48): X(cid:48) → X
extending g. By (c), there is an embedding f(cid:48): X(cid:48) → G that is identity on A. Finally,
f := f(cid:48)◦(g(cid:48))−1 is an embedding of X into G that is identity on A.
We shall say that G ∈ σF is weakly F-injective if it is F-universal and satisfies any of
the equivalent conditions of Proposition 3.1. Furthermore, we shall say that G is weakly
injective if it is weakly F-injective with F = Age(G).
Theorem 3.2. Suppose Eve does not have a winning strategy in BM(F,G). Then G is
weakly F-injective.
Proof. First of all, note that G is F-universal, since otherwise there would be A ∈ F not
embeddable into G and Eve would have a winning strategy, simply starting the game with
A := A. In order to show that G is not weakly F-injective, we shall use condition (b)
0
of Proposition 3.1. Namely, suppose (b) fails and fix a witness A (cid:54) G, A ∈ F. We shall
describe a winning strategy for Eve. Note that the following condition is fulfilled.
3
(×) For every isomorphism h: A(cid:48) → A, for every B (cid:62) A(cid:48) with B ∈ F, there exists
B(cid:48) (cid:62) B with B(cid:48) ∈ F such that no embedding of B(cid:48) into G extends h.
Eve starts with A := A. Suppose A (cid:54) A (cid:54) ... (cid:54) A are initial steps of the
0 0 1 n−1
game BM(F,G), where n is even. Eve chooses an isomorphism h whose domain is a
n
substructure of A and whose range is A. Then she responds with A := B(cid:48) from
n−1 n
condition (×) applied to h := h and B := A . By this way no embedding of A into G
n n−1 n
extendsh . ThisdescribesEve’sstrategy. Notethatateachsteptherearecountablymany
n
possibilities for choosing an isomorphism onto A, therefore an easy book-keeping makes
sure that Eve considers all of them. By this way she wins, as in the end no embedding of
(cid:83)
A into G can contain A in its image.
n∈ω n
Theorem 3.3. Suppose G ∈ σF is weakly F-injective. Then Odd has a winning strategy
in BM(F,G).
Proof. Let {v } enumerate a fixed set of generators of G. We shall use condition (c)
n n∈ω
of Proposition 3.1, knowing that G is F-universal.
Suppose A (cid:54) ... (cid:54) A form an initial part of BM(F,G) and n is odd. We assume
0 n−1
that on the way Odd had considered A(cid:48) ∈ F such that A (cid:54) A(cid:48) (cid:54) A for each even
i i i i+1
i < n−2, and he has recorded embeddings e : A(cid:48) → G such that e extends e and
i i i i−2
{v ,...,v } ⊆ e [A(cid:48)],
0 i i i
again for each even i < n−2. Furthermore, if n > 2, we assume that for every X ∈ F
with X (cid:62) A there exists an embedding e: X → G extending e . We now describe
n−2 n−3
Odd’s response.
Namely, OddfirstfindsacopyB (cid:54) GofA and, usingLemma2.3togetherwithour
n−1 n−1
inductive assumption, finds A(cid:48) (cid:62) A so that there is an embedding e : A(cid:48) → G
n−1 n−1 n−1 n−1
extending e (unless n = 1), whose image contains v for every i < n. Odd responds
n−3 i
with A (cid:62) A(cid:48) such that the assertion (c) of Proposition 3.1 holds with A := A(cid:48) ,
n n−1 n−1
e := e , and B := A . By this way, for every X ∈ F with X (cid:62) A , there is an
n−1 n n
embedding e: X → G extending e .
n−1
Using this strategy Odd in particular builds an embedding e : A → G, where A =
∞ ∞ ∞
(cid:83) A = (cid:83) A(cid:48) and e (cid:22) A(cid:48) = e for n ∈ 2N. Its image contains the set of
n∈ω n n∈2N n ∞ n n
generators {v } , therefore e is an isomorphism from A onto G.
n n∈ω ∞ ∞
4 Weak amalgamations
The following concept was introduced and used by Ivanov [5] and later by Kechris &
Rosendal [1]. Ivanov called it almost amalgamation property.
Definition 4.1. Let F be a class of finitely generated structures. We say that F has
the weak amalgamation property (briefly: WAP) if for every Z ∈ F there is Z(cid:48) ∈ F
containingZ asasubstructureandsuchthatforeveryembeddingsf: Z(cid:48) → X, g: Z(cid:48) → Y
with X,Y ∈ F there exist embeddings f(cid:48): X → V, g(cid:48): Y → V with V ∈ F, satisfying
f(cid:48)◦f (cid:22) Z = g(cid:48)◦g (cid:22) Z.
4
We also say that F has the cofinal amalgamation property (briefly: CAP) if
f(cid:48)◦f = g(cid:48)◦g
holds in the definition above. Finally, F has the amalgamation property (briefly: AP) if
Z(cid:48) = Z in the definition above.
A subclass F(cid:48) of a class F is called cofinal if F ⊆ F(cid:48)↓ .
One needs to admit here that the cofinal amalgamation property (which perhaps belongs
to the folklore) had been considered earlier by Truss [8]. Obviously, CAP implies WAP
and AP implies CAP. Note also that the CAP is equivalent to the existence of a cofinal
subclass with the AP. Finite cycle-free graphs provide an example of a hereditary class
satisfying CAP and not AP.
Proposition 4.2. Let F be a class of structures such that there exists G ∈ σF that is
weakly F-injective. Then F has the weak amalgamation property.
Proof. Fix Z ∈ F. We may assume that Z (cid:54) G. We shall use condition (a) of Propo-
sition 3.1. Namely, let Z(cid:48) ∈ F be such that Z (cid:54) Z(cid:48) (cid:54) G and (a) holds with A := Z,
B := Z(cid:48). Fix embeddings f: Z(cid:48) → X, g: Z(cid:48) → Y with X,Y ∈ F. Applying Proposi-
tion 3.1 twice, we obtain embeddings f(cid:48): X → G, g(cid:48): Y → G such that both f(cid:48) ◦f and
g(cid:48)◦g are identity on Z. In particular, f(cid:48)◦f (cid:22) Z = g(cid:48)◦g (cid:22) Z.
Definition 4.3. Let F be a class of countable finitely generated structures. We shall
say that F is a weak Fra¨ıss´e class if it has JEP, WAP, and contains countably many
isomorphic types.
Corollary 4.4. Let G be a countable structure, let F be a cofinal subclass of Age(G), and
assume Eve does not have a winning strategy in BM(F,G). Then F is a weak Fra¨ıss´e
class.
Proof. That F has JEP and contains countably many types is the statement of Proposi-
tion 2.1. The WAP follows directly from Theorem 3.2 and Proposition 4.2.
It remains to show that every weak Fra¨ıss´e class has its limit, that is, a suitable count-
ably generated structure with the weak extension property. According to the remark on
page 320 in [1], this can be “carried over without difficulty” adapting the Fra¨ıss´e theory
presented in the book of Hodges [3]. One cannot disagree with such a statement, however
we are not aware of any text where it has actually been done, therefore we present the
details in the next section.
5 Limits of weak Fra¨ıss´e classes
Let F be as above, Z ∈ F, and let Z(cid:48) ∈ F be such that Z (cid:54) Z(cid:48). We shall say that
Z(cid:48) is Z-good if it satisfies the assertion of Definition 4.1, namely, for every embeddings
f: Z(cid:48) → X, g: Z(cid:48) → Y with X,Y ∈ F there exist embeddings f(cid:48): X → V, g(cid:48): Y → V
5
with V ∈ F, satisfying f(cid:48)◦f (cid:22) Z = g(cid:48)◦g (cid:22) Z. Note that WAP says that for every Z ∈ F
there is Z(cid:48) ∈ F such that Z (cid:54) Z(cid:48) and Z(cid:48) is Z-good, while CAP means that for every
Z ∈ F thereisZ(cid:48) ∈ F suchthatZ (cid:54) Z(cid:48) andZ(cid:48) isZ(cid:48)-good. NotealsothatifZ (cid:54) Z(cid:48) (cid:54) Z(cid:48)(cid:48)
and Z(cid:48) is Z-good then so is Z(cid:48)(cid:48).
We are now ready to prove the main result of this section.
Theorem 5.1. Let F be a weak Fra¨ıss´e class. Then there exists a unique, up to iso-
morphisms, structure G ∈ σF that is weakly F-injective, and such that F is cofinal
in Age(G). Conversely, if G is a countable weakly injective structure then every cofinal
subclass of Age(G) is a weak Fra¨ıss´e class.
The structure G from the first statement will be called the limit of F.
Proof. Note that the second (“conversely”) part is the combination of Propositions 2.1
and 4.2. It remains to show the first part. Uniqueness follows from Proposition 2.2,
therefore it remains to show the existence. The construction will rely on the following
very simple fact (called the Rasiowa-Sikorski Lemma), well-known in forcing theory:
Claim 5.2. Given a partially ordered set P = (cid:104)P,(cid:54)(cid:105), given a countable family D of cofinal
subsets of P, there exists an increasing sequence F = {p : n ∈ N} such that D∩F (cid:54)= ∅
n
for every D ∈ D.
Recall that D is cofinal in P if for every p ∈ P there is q ∈ D such that p (cid:54) q.
Now, fix a weak Fra¨ıss´e class F of finitely generated structures. First, we “localize” F:
Namely, we assume that each A ∈ F lives in the set N of nonnegative integers and the
complement N\A is infinite. Define the following poset P. The universe of P is the class
F (refined as above) while (cid:54) is the usual relation of “being a substructure”. Define
E = {X ∈ F: C embeds into X},
C
where C ∈ F. By assumption, there are countably many sets of the form E and the
C
joint embedding property implies that each E is cofinal in P. Next, given A (cid:54) A(cid:48) in F
C
such that A(cid:48) is A-good, given an embedding f: A(cid:48) → B, define
D = {X ∈ F: If A(cid:48) (cid:54) X then (∃ an embedding g: B → X) (∀ x ∈ A) g(f(x)) = x}.
A,f
By assumption, there are countably many sets of the form D . Each of them is cofinal,
A,f
becauseoftheweakamalgamationproperty. Weonlyneedtorememberthatallstructures
in F are co-infinite in N, so that we always have enough space to enlarge them.
Finally, a sequence
U (cid:54) U (cid:54) U (cid:54) U (cid:54) ...
0 1 2 3
(cid:83)
obtained from Claim 5.2 to our family of cofinal sets produces a structure U = U
∞ n∈N n
in σF that is weakly F-injective. In particular, F is cofinal in Age(U ).
∞
One of the most important features of Fra¨ıss´e limits is universality. Namely, if F is a
Fra¨ıss´e class and G is its limit then every X ∈ σF embeds into G. This is not true for
weak Fra¨ıss´e classes (see Example 5.4 below), however the following weaker statement
holds true.
6
Theorem 5.3. Let F be a weak Fra¨ıss´e class and let G be its limit. Then for every chain
X (cid:54) X (cid:54) X (cid:54) ···
0 1 2
of structures in F such that X is X -good for each n ∈ ω, the union (cid:83) X embeds
n+1 n n∈ω n
into G.
Proof. Let us play the game BM(F,G) in such a way that Odd uses his winning strategy.
(cid:83)
We shall describe Eve’s strategy leading to an embedding of X = X into G.
n∈ω n
Eve starts with any A ∈ F for which there is an embedding e : X → A . Odd responds
0 1 1 0
withA (cid:62) A . EveusestheWAPinordertofindA (cid:62) A andanembeddinge : X → A
1 0 2 1 2 2 2
so that e (cid:22) X = e (cid:22) X . In general, when n = 2k and the position of the game is
2 0 1 0
A (cid:54) ... (cid:54) A , we assume that Eve has already recorded embeddings e : X → A
0 n−1 i i 2i−2
satisfying
((cid:63)) e (cid:22) X = e (cid:22) X
i+1 i−1 i i−1
for each i (cid:54) k. Eve responds with A = A (cid:62) A using the WAP, so that there is
n 2k n−1
an embedding e : X → A satisfying e (cid:22) X = e (cid:22) X . By this way, after
k k 2k k k−2 k−1 k−2
infinitely many steps of the game Eve has recorded embeddings e : X → A satisfying
i i 2i−2
((cid:63)) for every i ∈ ω. Define e = (cid:83) e (cid:22) X . Then e: X → (cid:83) A is a well-
i∈ω i i−1 n∈ω n
(cid:83)
defined embedding, because of ((cid:63)) and A ≈ G, because Odd was using his winning
n∈ω n
strategy.
Note that the above result gives the well-known universality of Fra¨ıss´e limits. This is
because if F is a Fra¨ıss´e class then every X ∈ F is X-good, therefore Theorem 5.3
applies to every structure in σF. Below is the announced example showing that the
result above cannot be improved.
Example 5.4. Let F be the class of all finite linear graphs, that is, finite graphs with
no cycles and of vertex degree (cid:54) 2. Connected graphs form a cofinal subclass with the
AP, therefore F is a weak Fra¨ıss´e class. Its limit is Z, the integers with the linear graph
structure (n is connected precisely to n+1 and n−1 for each n). The graph X consisting
of two disjoint copies of Z is in σF, however it cannot be embedded into Z.
Another important feature of Fra¨ıss´e limits is homogeneity, namely, every isomorphism
between finitely generated substructures extends to an automorphism. This is obviously
false in the case of weak Fra¨ıss´e classes (see Example 5.4 above: homogeneity totally fails
for disconnected subgraphs of Z). On the other hand, the following weaker result is true.
Theorem 5.5. Let F be a weak Fra¨ıss´e class with limit G. Then for every A (cid:54) A(cid:48) (cid:54) G
such that A,A(cid:48) ∈ F and A(cid:48) is A-good, for every embedding e: A(cid:48) → G there exists an
automorphism h: G → G such that h (cid:22) A = e (cid:22) A.
Proof. Theproofisasuitableadaptationoftheclassicalback-and-forthargument. Namely,
let A := A, A := A(cid:48), f := e. Let B (cid:54) G be in F, such that f [A ] ⊆ B . Choose
0 1 1 1 1 1 1
B ∈ F such that B (cid:54) B (cid:54) G and B is B -good. Applying the weak extension
2 1 2 2 1
property, find g : B → G such that g ◦ f (cid:22) A is identity. Let A ∈ F be such
2 2 2 1 0 2
7
that g [B ] ⊆ A (cid:54) G. Enlarging A if necessary, we may assume that it is A -good.
2 2 2 2 1
Inductively, we construct two chains of structures in F
A (cid:54) A (cid:54) A (cid:54) ... (cid:54) G and B (cid:54) B (cid:54) B (cid:54) ... (cid:54) G
0 1 2 1 2 3
and embeddings f : A → B and g : B → A satisfying for every n ∈ ω
2n+1 2n+1 2n+1 2n 2n 2n
the following conditions:
(1) A is A -good and B is B -good,
n+1 n n+1 n
(2) g ◦f is identity on A ,
2n 2n−1 2n−2
(3) f ◦g is identity on B ,
2n+1 2n 2n−1
(cid:83) (cid:83)
(4) A = G = B .
n∈ω n n∈ω n+1
Given f and g , we find g and f exactly in the same way as in the first step,
2n−1 2n−2 2n 2n+1
using the weak extension property of G. Note that we have a freedom to enlarge A
2n+2
and B as much as we wish, therefore we can easily achieve (4), knowing that G is
2n+1
countably generated.
Now observe that
f (cid:22) A = f ◦g ◦f (cid:22) A = f (cid:22) A ,
2n−1 2n−2 2n+1 2n 2n−1 2n−2 2n+1 2n−2
because f [A ] ⊆ B and hence we were able to apply (3) and (4). It follows
2n−1 2n−2 2n−1
that
(cid:91)
f := f (cid:22) A
∞ 2n+1 2n
n∈ω
is a well-defined embedding of G into itself. The same argument shows that
(cid:91)
g := g (cid:22) B
∞ 2n+2 2n+2
n∈ω
is a well-defined embedding of G into itself. Conditions (2), (3) make sure that f ◦g =
∞ ∞
id = g ◦f , showing that f is an isomorphism. Finally,
G ∞ ∞ ∞
f (cid:22) A = f (cid:22) A = f (cid:22) A = e (cid:22) A.
∞ ∞ 0 1 0
This completes the proof.
Note that, again, if F is a Fra¨ıss´e class then the result above gives full homogeneity,
namely, that every embedding between finitely generated substructures extends to an
automorphism.
6 Final remarks
Our results show that the weak amalgamation property plays the crucial role in the game.
Thus it is natural to ask the following:
8
Problem 6.1. Find a hereditary weak Fra¨ıss´e class without the cofinal amalgamation
property. How about weak Fra¨ıss´e classes of finite graphs?
Recently, Aristotelis Panagiotopoulos (private communication) has found an example of a
weak Fra¨ıss´e class of finite structures without the CAP. It remains open whether such a
class can be found among finite graphs.
9
7 Addendum
Firstofall,wewouldliketopointoutthatthefirstexampleofacountablehereditaryclass
of finite structures with JEP, WAP and without CAP was found by Alex Kruckman in
2015 (email communication). This is contained in [6, Example 3.4.7 on p. 73]. Formally,
this example is not hereditary, however its hereditary closure still fails the CAP.
Below is an example of a hereditary class of finite graphs having the JEP, WAP, and
lacking the CAP. We denote by deg (x) the degree of a vertex x in the graph G.
G
Example 7.1. Let G be the class of all finite cycle-free graphs in which no two vertices of
degree > 2 are adjacent. Obviously, G is hereditary and has the JEP. Furthermore, every
graph in G can be extended to a connected one, simply adding a new vertex connected
to selected vertices of degree 1 from each component. Being cycle-free ensures that each
G ∈ G has at least one vertex of degree (cid:54) 1.
We first check that G fails the CAP. Fix a non-discrete (i.e., containing at least one edge)
H ∈ G and choose v ∈ H of degree 1 (such a vertex exists, because there are no cycles).
Let X ⊇ H be such that X \H looks as
where v(cid:48) is adjacent to v in X and there are no further new adjacencies. Let Y ⊇ H be
such that X \H looks as
where v(cid:48)(cid:48) is adjacent to v in Y and there are no further new adjacencies in Y. The only
potential possibility of amalgamating these two embeddings in G would be by gluing v(cid:48)
and v(cid:48)(cid:48). But then z becomes a vertex adjacent to v(cid:48) and both of them have degree > 2, a
contradiction.
We now check that G has the WAP. Let us call H ∈ G tame if it satisfies the following
conditions:
(1) H is connected and has more than one vertex.
(2) Every vertex of degree 2 in H has a neighbor of degree > 2.
(3) The unique neighbor of every vertex of degree 1 in H has degree 2.
We claim that every member of G can be enlarged to a tame one. Fix H ∈ G. It is clear
how to enlarge it in order to satisfy (1) and (3). In order to achieve (2), fix a vertex v ∈ H
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