CONTINUUM MANY NON-ISOMORPHIC MODELS OF THE
THEORY OF A HRUSHOVSKI CONSTRUCTION FOR A
BINARY RELATION WITH AN IRRATIONAL COEFFICIENT
OMER MERMELSTEIN
1. Preliminaries
For the entirety of the text, fix some α ∈ (0, 1) irrational. We work in the
language of undirected graphs L = {R}.
For every finite structure (undirected graph) A define
δ(A) = |A| − α|R(A)|
and for B a superstructure of A, say A 6 B if δ(C) ≥ δ(A) for any structure
A ⊆ C ⊆ B. Let Kα to be the class of structures A with ∅ 6 A and let Kα to be
the class of finite structures of Kα .
The class (Kα , 6) is an (free) amalgamation class. The same is true of (Kα , 6)
with the proper notion of 6 for infinite sets.
Let Sα be the theory such that if M |= Sα then:
(1) Every substructure of M is an element of Kα .
(2) If A ⊆ M is finite, B ∈ Kα and A 6 B, then there is an embedding B ,→ M
over A.
In [Las07] it is shown that Sα is the complete theory of the Fraı̈ssé limit of
(Kα , 6).
2. Kα facts
Lemma 1. For every 1 > 2 > 0 there is some A ∈ Kα with 2 < δ(A) < 1 .
Proof. Left for later. But it really is quite believable.
Lemma 2. Let A ∈ Kα and let B ∈ Kα be such that A 6 B, then there is some
C ∈ Kα such that:
(i) B ⊆ C;
(ii) A 6 C;
(iii) p 66 C for every p ∈ (C \ A).
Proof. Let P = {p ∈ (B \ A) | p 6 B} and let β = δ(B/A). Let D ∈ Kα be such
that − |Pβ | < δ(D) − α < 0. Consider the structure C attained by attaching (by an
edge) a copy of D to each point in P . Then clearly (iii) holds in C and 0 < δ(C/A).
It remains to convince oneself that indeed A 6 C.
Date: August 7, 2016.
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OMER MERMELSTEIN
3. The construction of MB
Definition 3. For each n ∈ N let Wn be the structure which is a cycle of length
n. Formally,
Wn = {a1 , . . . , an }
R(Wn ) = {{ai , ai+1 } | 1 ≤ i ≤ n − 1} ∪ {{an , a1 }}.
For each D ⊆ N let WD be the free join
[
Wn
n∈B
Let mα ∈ N be such that mα (1 − α) > 1. Note that for every m ∈ N≥mα ,
every point p ∈ Wm is strong in Wm . Fix some D ⊆ N≥mα . We describe the
construction of a structure MD . This will be a tweaked version of the Fraı̈sséHrushovski construction.
Fix a countably infinite set M such that WD ⊆ M is coinfinite in M .
Let K ⊆ Kα be a set of representatives of the isomorphism types of Kα (i.e. for
every B ∈ Kα there is some A ∈ K with A ∼
= B).
Let F = {f : A → M | A ∈ K} be the family of possible mappings (as sets) of
structures from K into M .
Let T ⊆ F × K × K be the countable family of tuples defined defined
T = {(f, A, B) | A, B ∈ K, A 6 B, f ∈ F, dom(f ) = A}
and let {(fi , Ai , Bi )}i∈ω be a fixed enumeration of T such that each tuple of T is
repeated infinitely often. For each i ∈ ω, fix some Ci as guaranteed by Lemma 2
with respect to Ai 6 Bi .
We inductively construct an ascending chain of structures
WD = M0 ⊆ M1 ⊆ M2 ⊆ . . .
such that Mi ⊆ M , Mi ∈ Kα and Mi 6 Mi+1 for every i ∈ ω.
Assume that Mi is given. If Im(fi ) 66 Mi , define Mi+1 = Mi . Otherwise,
fi is a strong embedding of Ai into Mi . Without loss of generality, identify Ai
with fi (Ai ). Now, take Mi+1 to be the free join of Mi with Ci over Ai . Note
that Mi , Ci 6 Mi+1 . By renaming elements of Mi+1 \ Mi , we may assume that
Mi+1 ⊆ M .
S
Finally, let MD = i∈ω Mi .
Remark. The resulting structure MD is not unique and may depend on the choices
made during the construction. For our purposes, it is sufficient for MD to be some
structure constructed in this manner.
Lemma 4. MD |= Sα .
Proof. As MD is an ascending union of elements of Kα and so any of its substructures is an element of Kα .
Now, let A ⊆ MD be a finite substructure and let B ∈ Kα such that A 6 B.
Without loss of generality assume A, B ∈ K. Let j ∈ N be such that A ⊆ Mj .
Choose some stage i ∈ ω with i > j such that (fi , Ai , Bi ) = (IdA , A, B). Then Ci
embeds over A into Mi+1 ⊆ MD , and in particular B embeds into MD over A. CONTINUUM MANY NON-ISOMORPHIC MODELS OF THE THEORY OF A HRUSHOVSKI CONSTRUCTION FOR A BINARY R
Lemma 5. For each n ∈ N≥mα , Wn embeds strongly into MD if and only if
n ∈ D.
Proof. Firstly, assume n ∈ D. Then Wn 6 WD and, since WD = M0 6 MD , in
particular Wn 6 MD .
Now, assume Wn embeds strong into MD . Identify Wn with the image of this
embedding. Since n ≥ mα , for each p ∈ Wn we know that p 6 Wn and therefore,
by transitivity, p 6 MD . Let i ∈ ω be the smallest such that Wn 6 Mi+1 . By
choice of Ci , we know that if p ∈ (Ci \ Mi ), then p 66 Ci and in particular p 66 MD .
Thus, as every point of Wn is strongly embedded in MD , we have Wn ⊆ Mi . By
the minimality condition on i, this must mean Wn 6 M0 = WD . Then Wn embeds
into WD , and by construction of WD we have n ∈ D.
Theorem 6. There are continuum many non-isomorphic models of Sα .
Proof. By lemma 5, for every distinct D, D0 ⊆ N≥mα we have that MD MD0
since one of them strongly embeds a structure that the other does not. By lemma
4, MD is a model of Sα for every D ⊆ N≥mα . Thus, we have a collection of 2ℵ0
(the size of the powerset of N≥mα ) many non-isomorphic models of Sα .
References
[Las07] Michael C. Laskowski. A simpler axiomatization of the Shelah-Spencer almost sure theories. Israel J. Math., 161:157–186, 2007.