$\text{VC}_{\ell}$-dimension and the jump to the fastest speed of a hereditary $\mathcal{L}$-property

0.22 MB·

by  C. Terry

VC -dimension and the jump to the fastest speed of a hereditary
ℓ
L-property
C. Terry
7
1
0
Abstract
2
In this paper we investigate a connection between the growth rates of certain classes of
n
a finite structures and a generalization of VC-dimension called VCℓ-dimension. Let L be a finite
J relational language with maximum arity r. A hereditary L-property is a class of finite L-

2 structures closed under isomorphism and substructures. The speed of a hereditary L-property
H is the function which sends n to |Hn|, where Hn is the set of elements of H with universe
]
O {1,...,n}. It was previously known there exists a gap between the fastest possible speed of a
hereditary L-property and all lower speeds, namely between the speeds 2Θ(nr) and 2o(nr). We
L strengthen this gap by showing that for any hereditary L-property H, either |Hn| = 2Θ(nr) or
.
h thereisǫ>0suchthatforalllargeenoughn, |Hn|≤2nr−ǫ. Thisimproveswhatwaspreviously
t
a known about this gap when r ≥ 3. Further, we show this gap can be characterized in terms
m of VCℓ-dimension, thereforedrawinga connectionbetweenthis finite countingproblemandthe
[ model theoretic dividing line known as ℓ-dependence.

1
v 1 Introduction
0
7
4 One of the major themes in model theory is the search for dividing lines among first order theories.
0
The study of dividing lines was first developed by Shelah [16]. One of the main goals of this work
0
was to understand the function I(T,κ), which, given an input theory T and a cardinal κ, outputs
.
1
the number of non-isomorphic models of T of size κ. Therefore, the discovery of dividing lines
0
7 was fundamentally related to infinitary counting problems. Further, many dividing lines can be
1 characterized by a counting dichotomy, including stability, NIP, VC-minimality, and ℓ-dependence.
:
v These facts show us that model theoretic dividing lines are closely related to counting problems in
i
X the infinite setting.
r There has been substantial work on understanding dichotomies in finitary counting problems
a
in the field of combinatorics, particularly in the setting of graphs. A hereditary graph property is
a class of finite graphs H, which is closed under isomorphism and induced subgraphs. Given a
hereditary graph property, H, the speed of H is the function n 7→ |H |, whereH denotes the set of
n n
elements in H with vertex set [n]:= {1,...,n}. The possible speeds of hereditary graph properties
are well understood. In particular, their speeds fall into discrete growth classes, as summarized in
the following theorem.
Theorem 1. Suppose H is a hereditary graph property. Then one of the following holds, where
B ∼ (n/logn)n denotes the n-th Bell number.
n
1. Thereare rationalpolynomials p ,...,p suchthatforsufficientlylargen, |H |= k p (n)in,
0 k n Pi=0 i
2. There exists an integer k > 1 such that |Hn|= n(1−k1+o(1))n,
3. There is an ǫ > 0 such that for sufficiently large n, B ≤|H | ≤ 2n2−ǫ,
n n
1
4. There exists an integer k > 1 such that |Hn|= 2(1−k1+o(1))n2/2.
This theorem is the culmination of many authors’ work. We direct the reader to [4] for the gap
between cases 1 and 2 and within 2, to [4,6] for the gap between cases 2 and 3, to [2,9] for the gap
between 3 and 4, and to [9] for the gaps within case 4. Further, it was shown in [5] that there exist
hereditary graph properties whose speeds oscillate between the lower and upper bound of case 3,
therefore ruling out any more gaps in this range. Thus Theorem 1 solves the problem of what are
the possible speeds of hereditary graph properties.
On the other hand, there remain many open questions around generalizing Theorem 1, even to
thesettingofr-uniformhypergraphs,whenr ≥ 3. Wefocusononesuchprobleminthispaper. IfH
(n)
is a hereditary property of r-uniformhypergraphs,then |Hn| ≤ 2 r , and it was shown in [1]and [8]
that either |H | = 2cnr+o(nr) for some c > 0, or |H | ≤ 2o(nr). In other words, the fastest possible
n n
speed of a hereditary property of r-uniform hypergraphs is 2Θ(nr), and there is a gap between the
fastest and penultimate speeds. However, it remained openwhether this gap could bestrengthened
in analogy to the gap between cases 3 and 4 in Theorem 1, as we summarize below in Question 1.
Question 1. Suppose r ≥ 3. Is it true that for any hereditary property H of r-uniform hypergraphs,
either |H | =2cnr+o(nr) for some c> 0, or there is ǫ >0 such that for all large n, |H |≤ 2nr−ǫ?
n n
Given that model theoretic dividing lines are connected to infinitary counting problems, it is
natural to ask whether they are also connected to finitary counting problems such as Question 1.
The main results of this paper will establish such a connection, as well as answer Question 1 in the
affirmative.
Given a finite relational language L, a hereditary L-property is a class of finite L-structures, H,
closed under isomorphism such that if A is a model theoretic substructure of B and B ∈ H, then
A∈ H. The speed of H is the function n 7→ |H |, where H denotes the set of elements in H with
n n
universe [n]. The general problems we are interested in are the following.
• What are the jumps in speeds of hereditary L-properties?
• Can these jumps be characterized via model theoretic dividing lines?
In this paper, we make progress on these problem by improving the known the gap between the
penultimate and fastest possible speeds of a hereditary L-property, and by connecting this gap
to the model theoretic dividing line of ℓ-dependence. Specifically, we will characterize this gap
in terms of a cousin of VC-dimension, which we denote VC∗-dimension. We now state our main
ℓ
result, Theorem 2. We will then discuss how it improves known results and how it is connected to
ℓ-dependence.
Theorem 2. Suppose L is a finite relational language of maximum arity r ≥ 1, and H is a
hereditary L-property. Then either
(a) VC∗ (H) < ∞ and there is an ǫ > 0 such that for sufficiently large n, |H |≤ 2nr−ǫ, or
r−1 n
(b) VC∗ (H) = ∞, and there is a constant C > 0 such that |H | = 2Cnr+o(nr).
r−1 n
When r = 1, the following stronger version of (a) holds: VC∗(H) < ∞ and there K > 0 such that
0
for sufficiently large n, |H |≤ nK.
n
Theorem 2 strengthens what was previously shown in [22], that for any hereditary L-property
H, either |H | = 2Cnr+o(nr) for some C > 0, or |H | ≤ 2o(nr), where r is the maximum arity of the
n n
2
relations in L. This result generalizes the gap between cases 3 and 4 in Theorem 1, and is new in
all cases where r ≥ 3. Theorem 2 answers Question 1 in the affirmative.
Theorem 2 also shows the gap between the penultimate and fastest possible speeds of a hered-
itary L-property is characterized by a model theoretic dividing line. The dimension appearing
in Theorem 2, VC∗-dimension, is a dual version of the existing model theoretic notion of VC -
ℓ ℓ
dimension (see Section 2 for precise definitions). VC -dimension is a direct generalization of VC-
ℓ
dimension defined in terms of shattering “ℓ-dimensional boxes.” This dimension was first intro-
duced in [19], where it is used to define the dividing line called ℓ-dependence. VC -dimension and
ℓ
ℓ-dependencehave sincebeenstudiedfromthemodeltheoretic pointof view in[7,12,13,17,18]. We
will show that the condition VC∗(H) < ∞ is a natural analogue of ℓ-dependence for a hereditary
ℓ
L-property H. ThusTheorem 2 can beseen as characterizing agap in possiblespeedsof hereditary
L-properties using a version of the model theoretic dividing line of ℓ-dependence.
Our next result shows that the gap between polynomial and exponential growth is always
characterized by VC∗-dimension, regardless of the arity of the language.
0
Theorem 3. Suppose L is a finite relational language, and H is a hereditary L-property. Then
either
(a) VC∗(H) < ∞ and there K > 0 such that for sufficiently large n, |H |≤ nK, or
0 n
(b) VC∗(H) = ∞, and there is a constant C > 0 such that for sufficiently large n, |H | ≥ 2Cn.
0 n
Theorem3isnewatthislevel ofgenerality, inthelabeledsetting. Thereexistgeneralresultson
the polynomial/exponential counting dichotomy in the unlabelled setting (see for instance [14,15]),
anditispossiblethemachinerydevelopedinthatlineofworkcouldbeusedtoobtainthedichotomy
of Theorem 3. The connection this paper makes between this problem and VC -dimension is
ℓ
new. Thus, while the existence of the dichotomy described by Theorem 3 is not surprising given
past results, Theorem 3 draws a connection to VC -dimension which we think is important for
ℓ
understanding the larger pattern at work.
Thedichotomies in Theorems 2 and 3 dependon whether VC -dimension is finiteor infinite, for
ℓ
certainvaluesofℓ. Bothresultsusethefollowingtheorem,whichshowsthatinfiniteVC∗-dimension
ℓ
always implies a lower bound on the speed.
Theorem 4. Suppose L is a finite relational language of maximum arity r, and H is a hereditary
L-property. If1≤ ℓ ≤ r andVC∗ (H)= ∞, thenthere isC > 0suchthatforlargen, |H | ≥ 2Cnℓ.
ℓ−1 n
Somewhat surprisingly, the converse of Theorem 4 fails. In particular, we will give an example
of a hereditary property of 3-uniform hypergraphs with VC (H) < ∞ but with |H | ≥ 2Cn2 for
1 n
some C > 0 (see Example 1). We would like to thank D. Mubayi for bringing said example to our
attention. These observations suggest the following interesting open problem.
Problem 1. Suppose L is a finite relational language of maximum arity r ≥ 3 and ℓ is an integer
satisfying 2 ≤ ℓ < r. Say a hereditary L-property H has fast ℓ-dimensional growth if |H | ≥ 2Ω(nℓ).
n
Characterize the hereditary L-properties with fast ℓ-dimensional growth.
We end this introduction with a brief outline of the paper. In Section 2 we give background
on VC -dimension and VC∗-dimension. In Section 3 we present technical lemmas needed for the
ℓ ℓ
proofs of our main results. In Section 4 we prove Theorems 2 and 3 and present Example 1. In
Section 5, we prove that when ℓ > 0, VC∗(H) = ∞ if and only if VC (H) = ∞.
ℓ ℓ
3
2 Preliminaries
In this section, we introduce VC -dimension for ℓ ≥ 1 and VC∗-dimension for ℓ ≥ 0. For this
ℓ ℓ
section, Lissomefixedlanguage. We willdenoteL-structureswithscriptletters, e.g. M, andtheir
universes with the corresponding non-script letters, e.g. M. Given an integer n, [n] := {1,...,n}.
If X is a set, X = {Y ⊆ X : |Y|= n}, and if x¯ = (x ,...,x ) is a tuple, then |x¯| = s.
(cid:0)n(cid:1) 1 s
2.1 VC-dimension and VC -dimension
ℓ
In this subsection we define VC-dimension and VC -dimension. We begin by introducing VC-
ℓ
dimension. Given sets A⊆ X, P(X) denotes the power set of X. If F ⊆ P(X), then F∩A denotes
the set {F ∩A : F ∈ F}. We say A is shattered by F if F ∩A = P(A). The VC-dimension of
F is VC(F) = sup{|A| : A ⊆ X is shattered by F}, and the shatter function of F is defined by
π(F,m) = max{|F ∩A| : A ∈ X }. Observe that VC(F) ≥ m if and only if π(F,m) = 2m. One
(cid:0)m(cid:1)
of the most important facts about VC-dimension is the Sauer-Shelah Lemma.
Theorem 5 (Sauer-Shelah Lemma). Suppose X is a set and F ⊆ P(X). If VC(F) = d, then
there is a constant C = C(d) such that for all m, π(F,m) ≤ Cmd.
VC-dimension is important in various fields, including combinatorics, computer science, and
model theory. We direct the reader to [20] for more details. Given ℓ ≥ 1, VC -dimension is a
ℓ
generalization of VC-dimension whichfocuseson theshattering sets ofaspecial form. IfX ,...,X
1 ℓ
are sets, then ℓ X is an ℓ-box. If |X | = ... = |X | = m, then we say ℓ X is an ℓ-box of
Qi=1 i 1 ℓ Qi=1 i
height m. If X′ ⊆ X ,...,X′ ⊆ X , then ℓ X′ is a sub-box of ℓ X .
1 1 ℓ ℓ Qi=1 i Qi=1 i
Definition 1. Suppose ℓ ≥ 1, ℓ X is an ℓ-box, and F ⊆ P( ℓ X ). The VC -dimension of
Qi=1 i Qi=1 i ℓ
F is
ℓ
VC (F) = sup{m ∈ N : F shatters a sub-box of X of height m}.
ℓ Y i
i=1
Theℓ-dimensionalshatterfunction isπ (F,m) = {|F∩A| :A is a sub-box of ℓ X of height m}.
ℓ Qi=1 i
VC -dimensionwasintroducedinthemodeltheoreticcontextin[19],whereitwasusedtodefine
ℓ
the notion of an ℓ-dependent theory. It has since been studied as a diving line in [7,12,13,17,18].
Theorem6, below, isananalogueof theSauer-ShelahLemmaforVC -dimension,whichwas proved
ℓ
in [12].
Theorem 6 (Chernikov-Palacin-Takeuchi [12]). Suppose ℓ ≥ 1, Y is an ℓ-box, and F ⊆ P(Y).
If VC (F) = d < ∞, then there are constants C = C(d) and ǫ = ǫ(d) > 0 such that for all m ∈ N,
ℓ
π (F,m) ≤ C2mℓ−ǫ.
ℓ
We willneedmorecomplicated versionsof Definition 1andTheorem6. Thisextracomplication
comesfromthefactthatforthispaper,wecannotworkinsideTeq,asisdonein[12](wewillnoteven
beworkinginacompletetheory). Wenowfixsomenotation. SupposeX isaset,andk ,...,k ≥ 1
1 ℓ
are integers. Given a¯1 ∈ Xk1,...,a¯ℓ ∈ Xkℓ, let a¯1...a¯ℓ denote the element of Xk1+...+kℓ which
is the concatenation of the tuples a¯1,...,a¯ℓ. Given nonempty sets A1 ⊆ Xk1,...,Aℓ ⊆ Xkℓ, let
A ...A := {a¯ ...a¯ : a¯ ∈ A ,...,a¯ ∈ A }. Abusing notation slightly, we will write ℓ A for
1 ℓ 1 ℓ 1 1 ℓ ℓ Qi=1 i
the set A ...A . Observe ℓ A ⊆ Xr, where r = k +...+k . We call ℓ A an (ℓ,r)-box in
1 ℓ Qi=1 i 1 ℓ Qi=1 i
X. If |A | = ... = |A | = m for some m ∈ N, then we say ℓ A has height m. By convention,
1 ℓ Qi=1 i
for r ≥ 1, a (0,r)-box of any height in X is a singleton in Xr, and a (0,0)-box of any height in X
is the empty set. Given any 0 ≤ ℓ ≤ r, we will say a set A is an (ℓ,r)-box if there is some set X
such that A is an (ℓ,r)-box in X.
4
Definition 2. Suppose X is a set, 1≤ ℓ ≤ r, and F ⊆ P(Xr). The VC -dimension of F is
ℓ
VC (F) = sup{m ∈ N : F shatters an (ℓ,r)-box of height m in X}.
ℓ
The ℓ-dimensional shatter function is π (F,m) = {|F ∩A| :A is an (ℓ,r)-box in X of height m}.
ℓ
Theorem 6 can be directly adapted to these definitions.
Theorem 7. Suppose 1 ≤ ℓ ≤ r, X is a set and F ⊆ P(Xr). If VC (F) = d < ω, then there are
ℓ
constants C = C(d) and ǫ = ǫ(d) > 0 such that for all m, π (F,m) ≤ C2mℓ−ǫ.
ℓ
Proof. Observe that any (ℓ,r)-box in X is a sub-box of Qℓi=1Xki, for some k1,...,kℓ ≥ 1 with
k1+...+kℓ = r. Given k1,...,kℓ ≥ 1 such that k1+...+kℓ = r, let F(k1,...,kℓ) = F∩Qℓi=1Xki.
Our observation implies that F shatters an (ℓ,r)-box of height m in X if and only if F(k ,...,k )
1 ℓ
shatters a sub-box of Qℓi=1Xki of height m, for some k1,...,kℓ ≥ 1 with k1 + ... + kℓ = r.
Consequently,
π (F,m) = max{π (F(k ,...,k ),m) :k ,...,k ≥ 1,k +...+k = r} and (1)
ℓ ℓ 1 ℓ 1 ℓ 1 ℓ
VC (F) = max{VC (F(k ,...,k )) : k ,...,k ≥ 1,k +...+k = r}, (2)
ℓ ℓ 1 ℓ 1 ℓ 1 ℓ
where the left-hand sides are computed as in Definition 2 and the right-hand sides are computed
as in Definition 1. By assumption, VC (F) ≤ d, so (2) implies that for all k ,...,k ≥ 1 with
ℓ 1 ℓ
k + ... + k = r, VC (F(k ,...,k )) ≤ d. Therefore, by Theorem 6, there are C = C(d) and
1 ℓ ℓ 1 ℓ
ǫ = ǫ(d) > 0 such that for all m, π (F(k ,...,k ),m) ≤ C2mℓ−ǫ. Combining this with (1) implies
ℓ 1 ℓ
π (F,m) ≤ C2mℓ−ǫ holds for all m.
ℓ
Note that VC -dimension is the same as VC-dimension. Observe that in the notation of Def-
1
inition 2, for all m, π (F,m) ≤ 2mℓ, and VC (F) ≥ m if and only if π (F,m) = 2mℓ. We will be
ℓ ℓ ℓ
particularlyinterestedintheVC -dimensionoffamiliesofsetsdefinedbyformulasinanL-structure.
ℓ
Given a formula ϕ(x¯;y¯), an L-structure M, and ¯b ∈ M|x¯|, let
ϕ(¯b;M) = {a¯ ∈ M|y¯| :M |= ϕ(¯b;a¯)} and F (M) = {ϕ(¯b;M) :¯b ∈ M|x¯|}.
ϕ
Note F (M) ⊆ P(M|y¯|). If A ⊆ M|y¯|, we say ϕ shatters A if F (M) does. Given 1 ≤ ℓ ≤ |y¯|, set
ϕ ϕ
VC (ϕ,M) = VC (F (M)). Then if H is a hereditary L-property, set
ℓ ℓ ϕ
VC (ϕ,H) =sup{VC (ϕ,M) : M∈ H}.
ℓ ℓ
We now define the VC -dimension of a hereditary L-property, for ℓ ≥ 1.
ℓ
Definition 3. Suppose ℓ ≥ 1, and H is a hereditary L-property. Then
VC (H) = sup{VC (ϕ,H) :ϕ(x¯;y¯) ∈ L is quantifier-free},
ℓ ℓ
and we say H is ℓ-dependent if for all quantifier-free formulas ϕ(x¯;y¯), VC (ϕ,H) < ω.
ℓ
NotethatinDefinition3,wedefineVC (H)intermsofVC (ϕ,H)forquantifier-free ϕ. Because
ℓ ℓ
we are dealing with classes of finite structures, this turnsout to bethe appropriatenotion. We now
explain how this is related to the VC -dimension of a complete first-order theory and the notion of
ℓ
ℓ-dependence. Suppose T is a complete L-theory. Given a formula, ϕ(x¯;y¯), the VC -dimension of
ℓ
ϕ in T is VC (ϕ,T) := VC (ϕ,M), where M is a monster model of T and VC (ϕ,M) is computed
ℓ ℓ ℓ
precisely as described above. The theory T is ℓ-dependent if VC (ϕ,T) < ω for all ϕ ∈ L. This can
ℓ
5
be related to Definition 3 as follows. Let H(T) be the age of M (i.e. the class of finite L-structures
which embed into M). Then for any quantifier-free ϕ, VC (ϕ,H(T)) = VC (ϕ,T). Clearly if T
ℓ ℓ
is ℓ-dependent, then so is H(T). However, the converse will not hold if all quantifier-free formulas
have finiteVC -dimension in T, butthereis aϕ withquantifiers such thatVC (ϕ,T) = ω. Further,
ℓ ℓ
many hereditary L-propertiesarenotages (recall thatif Lis finiteandrelational, then ahereditary
L-property is an age if and only if it has the joint embedding property [11]). Thus, while one can
view Definition 3 as a version of ℓ-dependence adapted to the setting of hereditary L-properties, it
differs in fundamental ways from the notion of the VC -dimension of a complete theory.
ℓ
2.2 VC∗-dimension
ℓ
In this subsection we define VC∗-dimension, a dual version of VC -dimension. This is necessary
ℓ ℓ
because directly generalizing VC -dimension to the case when ℓ = 0 does not give us a useful
ℓ
notion. Indeed, for any formula ϕ(x¯) and L-structure M, ϕ trivially shatters a (0,0)-box (i.e. the
empty set). We would like to point out that VC∗-dimension is stronger than the dual version of
ℓ
VC -dimension appearing in [12].
ℓ
We now fix some notation. Suppose ϕ(x¯;y¯) is a formula, X is a set, and A ⊆ X|y¯|. A ϕ-type
over A in the variables x¯ is a maximal consistent subset of {ϕ(x¯;a¯)i : a¯ ∈ A,i ∈ {0,1}} (where
ϕ0 = ϕ and ϕ1 = ¬ϕ). Given an integer n, S∅(A) is the set of complete types in the language
n
of equality, using n variables, and with parameters in A. Given p in S (A) or S∅(A), we say p is
ϕ n
realized in an L-structure M if A ⊆ M|y¯|, and there is a¯ ∈ M|x¯| such that M |= p(a¯). If H is
a hereditary L-property, SH(A) is the set of complete ϕ-types over A which are realized in some
ϕ
M∈ H.
Definition 4. Suppose H is a hereditary L-property, m ≥ 1, ϕ(x¯;y¯) is a formula, X is a set, and
A ⊆ X|y¯|. Then SH (A) is the set of all ϕ-types of the form p (x¯ )∪...∪p (x¯ ), satisfying
ϕ,m 1 1 m m
1. For each i ∈[m], p (x¯ )∈ SH(A), and
i i ϕ
2. ThereisM∈ Handpairwise distinct a¯ ,...,a¯ ∈ M|x¯| suchthatM|= p (a¯ )∪...∪p (a¯ ).
1 m 1 1 m m
Givenρ∈ S∅ (A),SH (A,ρ)isthesetofp (x¯ )∪...∪p (x¯ )∈ SH (A)suchthatthereisM∈ H
2|x¯| ϕ,m 1 1 m m ϕ,m
and pairwise distinct a¯ ,...,a¯ ∈ M|x¯| with M|= p (a¯ )∪...∪p (a¯ )∪ ρ(a¯ ,a¯ ).
1 m 1 1 m m S1≤i6=j≤m i j
Observe that in the notation of Definition 4, for any (ℓ,|y¯|)-box A of height m, |SH(A)| ≤ 2mℓ and
ϕ
for all ρ ∈ S∅ (A), |SH (A,ρ)| ≤ |SH(A)|m. Consequently, |SH (A,ρ)| ≤ 2mℓ+1. We are now
2|x¯| ϕ,m ϕ ϕ,m
ready to define the VC∗-dimension of a hereditary L-property, for ℓ ≥ 0.
ℓ
Definition 5. Suppose ϕ(x¯;y¯) is a formula, H is a hereditary L-property, and 0 ≤ ℓ ≤ |y¯|. Then
VC∗(ϕ,H) = sup{m ∈N : for some (ℓ,|y¯|)-box A of height m and ρ ∈S∅ (A), |SH (A,ρ)| = 2mℓ+1},
ℓ 2|x¯| ϕ,m
and VC∗(H)= sup{VC∗(ϕ,H) :ϕ(x¯;y¯) ∈ L is quantifier-free}.
ℓ ℓ
Throughout we will use the notation VC∗(H) = ∞ instead of VC∗(H) = ω (and similarly for
ℓ ℓ
other dimensions). We will frequently use the following observation.
Observation 1. For all ℓ ≥ 0 and formulas ϕ(x¯;y¯), VC∗(ϕ,H) ≥ m if and only if there is an
ℓ
(ℓ,|y¯|)-box A of height m and ρ ∈ S∅ (A) such that |SH(A)| = 2mℓ and for all (p ,...,p ) in
2|x¯| ϕ 1 m
SH(A)m, p (x¯ )∪...∪p (x¯ ) ∈ SH (A,ρ).
ϕ 1 1 m m ϕ,m
6
On the other hand, note that for all ℓ > 0 and formulas ϕ(x¯;y¯), VC (ϕ,H) ≥ m if and only
ℓ
if there is an (ℓ,|y¯|)-box A of height m such that |SH(A)| = 2mℓ. Therefore VC∗(ϕ,H) ≥ m is a
ϕ ℓ
stronger statement than VC (ϕ,H) ≥ m.
ℓ
We now make a few remarks on our choice of definitions. We defined VC∗-dimension using
ℓ
SH (A,ρ) for ρ ∈ S∅ (A) in order to avoid pathologies in the case when ℓ = 0. In particular,
ϕ,m 2|x¯|
for any non-trivial hereditary L-property H with H 6= ∅, |SH (∅)| = 2n. Indeed, H 6= ∅
2n x=y,n 2n
implies that for any σ ∈ {0,1}n, SH (∅) contains {(x = y )σ(i) : 1 ≤ i ≤ n}. Therefore if
x=y,n i i
we defined VC∗-dimension using |SH (A)| instead of |SH (A,ρ)| for some ρ ∈ S∅ (A), every
0 ϕ,m ϕ,m 2|x¯|
hereditary L-property of interest to us would satisfy VC∗(x = x ,H) = ∞. Our definition avoids
0 1 2
this undesirable behavior when ℓ = 0. Further, we will prove in Section 5 that for any hereditary
L-property H and ℓ > 0, VC∗(H)= ∞ if and only if VC (H)= ∞. In light of this, we may extend
ℓ ℓ
Definition 3 to all ℓ ≥ 0 by saying a hereditary L-property H is ℓ-dependent if VC∗(ϕ,H) <∞ for
ℓ
all quantifier-free ϕ.
3 Technical Lemmas
In this section we present two technical lemmas which we will use in the proofs of our main results.
Since we are interested in counting, it is often important to distinguish between tuples and their
underlying sets. For this reason we will often denote sets of tuples using bold face letters, and
the corresponding underlying sets using non-bold letters. Objects which are tuples will always
have bars over them. For the rest of the paper L is a fixed finite relational language
with maximum arity r ≥ 1, and H is a hereditary L-property. For the rest of the paper,
“formula” always means quantifier-free formula. Since H is now fixed, we will from here on omit
the superscripts H from the notation defined in Definition 4.
ThefirstresultofthissectionisLemma1below. Parts(a)and(b)ofLemma1givequantitative
bounds for the size of indiscernible sets in the language of equality, and part (c) of Lemma 1 is
an easy but useful counting fact. The proof of Lemma 1 is straightforward and appears in the
appendix.
Lemma 1. Suppose X is a set, s,t ∈ N, B ⊆ Xt is finite, and B is the underlying set of B. Then
the following hold.
(a) |Ss∅(B)| ≤ 2(2s)(|B|+1)s.
(b) There is B′ ⊆ B which is an indiscernible subset of Xt in the language of equality satisfying
|B′|≥ (cid:16)|B|/2(2t)(cid:17)1/2t.
(c) |B|≤ t|B| and |B|1/t ≤ |B|.
If 0 < ℓ ≤ r and A = ℓ A is an (ℓ,r)-box, then a sub-box of A is an (ℓ,r)-box of the form
Qi=1 i
ℓ A′ where for each 1 ≤ i ≤ ℓ, A′ ⊆ A is nonempty. By convention, for any r ≥ 0, the only
Qi=1 i i i
sub-box of a (0,r)-box is itself. Our next result of this section is Lemma 2 below, which gives us
information about types over sub-boxes.
Lemma 2. Let ϕ(x¯;y¯) be a formula, and let ℓ,K,N,m be integers satisfying K >> N ≥ m ≥ 1,
and 0 ≤ ℓ. If A is an (ℓ,|y¯|)-box of height K satisfying |S (A)| = 2Kℓ, then for any sub-box A′ ⊆ A
ϕ
of height m, the following hold.
(a) The underlying set of A′ has size at most |y¯|m.
7
(b) |S (A′)| = 2mℓ.
ϕ
(c) Suppose ℓ > 0, M is an L-structure, and D ⊆ M|x¯| contains one realization of every element
of S (A). Then D contains at least N realizations of every element of S (A′).
ϕ ϕ
(d) If |S (A,ρ)| = 2Kℓ+1 for some ρ ∈ S∅ (A), then |S (A′,ρ↾ )| = 2mℓ+1, and there is
ϕ,K 2|x¯| ϕ,m A′
M∈ H and D ⊆ Mm|x¯| such that D contains one realization of every element of S (A′,ρ↾ ),
ϕ,m A′
and M contains at least N elements not in A′ or in any element of D.
Proof. Let A be the underlying set of A and let A′ be the underlying set of A′. We first show (a).
If ℓ = 0, then A = A′ implies either |y¯| = 0 and |A′| = 0 ≤ |y¯|m, or |y¯| > 0 and |A′| = 1 ≤ |y¯|m. If
ℓ > 0, then A = Qℓi=1Ai where for each i, Ai ⊆ Aki for some ki ≥ 1 and such that Pℓi=1ki = |y¯|.
Because A′ is a sub-box of A of height m, we have A′ = ℓ A′, where for each i, A′ ⊆ A has
Qi=1 i i i
size m. For each i, Lemma 1 part (c) implies |A′| ≤ k m. Consequently, |A′| ≤ ℓ k m = |y¯|m.
i i Pi=1 i
Thus (a) holds. For parts (b), (c), and (d), we will use the following claim.
Claim 1. There are Γ ,...,Γ ⊆ S (A) and pairwise distinct p ,...,p in S (A′) such that:
1 2mℓ ϕ 1 2mℓ ϕ
(i) For each 1≤ i ≤ 2mℓ, every element of Γ is an extension of p to A.
i i
(ii) If ℓ > 0, then |Γ | ≥ N.
i
Proof. Suppose first ℓ = 0. Then A′ = A. By assumption |S (A)|= 2Kℓ = 2. Let p ,p be the two
ϕ 1 2
distinct elements of S (A), and set Γ = {p }, and Γ = {p }. Then it is clear p 6= p ∈ S (A′)
ϕ 1 1 2 2 1 2 ϕ
and (i), (ii) hold. Suppose now ℓ ≥ 1. Let X ,...,X enumerate all the subsets of A′, and for
1 2mℓ
each 1≤ j ≤ 2mℓ, set p (x¯) = {ϕ(x¯;a¯): a¯ ∈ X }∪{¬ϕ(x¯;a¯): a¯ ∈ A′\X}. Given X ⊆ A\A′, and
j j
1 ≤ j ≤ 2mℓ, set
p (x¯) = {ϕ(x¯;a¯): a¯ ∈ X ∪X}∪{¬ϕ(x¯;a¯): a¯ ∈ A\(X ∪X)} and Γ = {p (x¯) :X ⊆ A\A′}.
j,X j j j j,X
Since |S (A)| = 2Kℓ, we must have that for all X ⊆ A, {ϕ(x¯;a¯) : a¯ ∈ X}∪{¬ϕ(x¯;a¯) : a¯ ∈ A\X}
ϕ
is in S (A). Consequently, for each 1 ≤ j ≤ 2mℓ, Γ ⊆ S (A). By definition, for all p ∈ Γ ,
ϕ j ϕ j
p↾ = p . For each j, since Γ ⊆ S (A), and since any realization of an element of Γ is a
A′ j j ϕ j
realization of p , we have p ∈ S (A′). By definition, p ,...,p are pairwise distinct. Thus
j j ϕ 1 2mℓ
we have shown p ,...,p are pairwise distinct elements of S (A′) and (i) holds. For each j,
1 2mℓ ϕ
|Γ |≥ |P(A\A′)| ≥ 2Kℓ −2mℓ ≥ N, where the last inequality is because K >> N ≥ m. Thus (ii)
j
holds. This finishes the proof of Claim 1. (cid:4)
Now fix Γ ,...,Γ ,p ,...,p as in Claim 1. Since the p are pairwise distinct elements of
1 2mℓ 1 2mℓ i
S (A′), we immediately have that |S (A′)| = 2mℓ, so (b) holds. We now show (c) holds. Suppose
ϕ ϕ
ℓ > 0, M is an L-structure, and D ⊆ M|x¯| contains one realization of every element of S (A). Then
ϕ
M contains a realization of every element in 2mℓ Γ . Since each Γ contains at least N extensions
Si=1 i i
of p , this shows M contains at least N realizations of each p . This finishes the proof of (c).
i i
We now prove (d). Suppose |S (A,ρ)| = 2Kℓ+1 for some ρ ∈ S∅ (A). Since K >> m, we
ϕ,K 2|x¯|
may assume K ≥ m2mℓ+1. Thus we may fix a sequence (α ,...,α ) ∈ S (A)K such that for each
1 K ϕ
1 ≤ j ≤ 2mℓ,
|{α ,...,α }∩Γ |= m. (3)
1 m2mℓ j
8
Then |S (A,ρ)| = 2Kℓ+1 implies by Observation 1 that α:= α (x¯ )∪...∪α (x¯ )∈ S (A,ρ).
ϕ,K 1 1 K K ϕ,K
Thus there is M ∈ H containing pairwise distinct a¯ ,...,a¯ realizing α such that for each i 6= j,
1 K
M |= ρ(a¯ ,a¯ ). For each 1 ≤ j ≤ 2mℓ, since every element of Γ extends p , (3) implies that
i j j j
{a¯ ,...,a¯ } contains m realizations of p . This means that for all (p ,...,p ) in S (A′)m,
1 m2mℓ j j1 jm ϕ
we may choose pairwise distinct tuples c¯ ,...,c¯ ∈ {a¯ ,...,a¯ } with the property that M |=
1 m 1 m2mℓ
p (c¯ ) ∪ ... ∪ p (c¯ ). Let D consist of one such realization for each (p ,...,p ) ∈ S (A′)m.
j1 1 jm m j1 jm ϕ
Note that (c¯ ,...,c¯ )∈ D implies M|= ρ↾ (c¯,c¯ ) for each i 6= j (since D ⊆{a¯ ,...,a¯ }m).
1 m A′ i j 1 m2mℓ
We have now shown that for every (p ,...,p ) ∈ S (A′)m, p (x¯ ) ∪ ... ∪ p (x¯ ) is in
j1 jm ϕ j1 1 jm m
Sϕ,m(A′,ρ↾A′). To finishthe proof of (d), we justhave to show that M contains at least N elements
not appearing in A′ or D. Let D be the underlying set of D and let E be the underlying set of the
tuples E = {a¯ ,...,a¯ }. Since K ≥ m2mℓ+1, |E| ≥ K/2. This along with Lemma 1 part (c)
m2mℓ+1 K
and the fact that E ⊆ M|x¯| implies (K/2)1/|x¯| ≤ |E|1/|x¯| ≤ |E|. Since D ⊆ M|x¯|m and |D| = m2mℓ,
Lemma 1 part (c) implies |D| ≤ |x¯|m2mℓ. We have already shown |A′| ≤ |y¯|m. Combining these
bounds,we obtain that |E\(A′∪D)|≥ (K/2)1/|x¯|−|x¯|m2mℓ−|y¯|m ≥N, where the last inequality
is because K >>N ≥ m. This finishes the proof of (d).
4 Proofs of Main Theorems
Inthissection, weprove themain resultsof thispaper. Webegin withTheorem4, which werestate
here for convenience. If M is an L-structure and A ⊆ M, then M[A] denotes the L-structure
induced on A by M.
Theorem 4. If 1 ≤ ℓ and VC∗ (H) = ∞, then there is C > 0 such that for sufficiently large n,
ℓ−1
|H | ≥2Cnℓ.
n
Proof. Assume 1 ≤ ℓ and VC∗ (H) = ∞. By definition, there is a formula ϕ(x¯;y¯) such that
ℓ−1
VC∗ (ϕ,H) = ∞. Let s = |x¯| and t = |y¯|+|x¯|. Fix n large and K >>n. Then VC∗ (ϕ,H) ≥ K
ℓ−1 ℓ−1
implies there is an (ℓ−1,|y¯|)-box A of height K and ρ∈ S∅ (A) such that |S (A,ρ)| = 2Kℓ. Fix
2s ϕ,K
m = ⌊n/3t⌋. Note m ≤ n<< K.
Choose a sub-box A′ ⊆ A of height m and let A′ be the underlying set of A′. By Lemma 2
part (a), |A′| ≤ |y¯|m ≤ tk. By Lemma 2 part (d), |Sϕ,m(A′,ρ↾A′)| =2mℓ, and there is M∈ H, and
D ⊆ Mm|x¯| such that D contains one realization of every element of S (A′,ρ↾ ), and M contains
ϕ,m A′
n elements not appearing in A′ or in D. Let D be the underlying set of D and let E ⊆ M be a set
of n elements in M \(A′∪D).
Given C¯ ∈ D, let C be the underlying set of C¯. For all C¯ ∈ D, C¯ ∈ Mm|x¯| implies by Lemma
1 part (c) that |C|≤ |x¯|m ≤ tm. Since every element of D realizes the same equality type over A′,
we have that for all C¯,C¯′ ∈ D, |C|= |C′| and |C ∪A′|= |C ∪A′|. Given C¯ ∈ D, note
|C ∪A′| ≤ |C|+|A′|≤ tm+tm = 2tm ≤ 2t(n/3t) = 2n/3. (4)
Since E has size n and is disjoint from D∪A′, (4) implies we may choose E′ ⊆ E such that for all
C¯ ∈ D,|C∪A′∪E′|= n. Now foreach C¯ ∈ D,setM = M[C∪A′∪E′]. Because H isahereditary
C¯
L-property, M ∈ H for all C¯ ∈ D. Fix some C¯ = (c¯∗,...,c¯∗ ) ∈ D. Given C¯ = (c¯ ,...,c¯ ) ∈ D,
C¯ ∗ 1 m 1 m
note that C¯ and C¯ have the same equality type over A′ ∪ E′. Therefore there is a bijection
∗
f : C ∪A′∪E′ → C ∪A′∪E′, which fixes A′∪E′ and which sends c¯ to c¯∗ for each 1 ≤ i ≤ m.
C¯ ∗ i i
Let M∗ be the L-structure with universe C ∪A′ ∪E′, and which is isomorphic to M via the
C¯ ∗ C¯
bijection f . Since H is closed under isomorphism, M∗ ∈ H for all C¯ ∈ D. Clearly C¯ 6= C¯′ implies
C¯ C¯
M∗ 6= M∗ (since then C¯ and C¯′ realize distinct elements of S (A,ρ↾ )). Thus {M∗ :C¯ ∈ D}
C¯ C¯′ ϕ,m A′ C¯
9
consists of |D| distinct elements of H, all with universe C ∪A′∪E′. Since |C ∪A′∪E′| = n and
∗ ∗
H is closed under isomorphism, this shows |H | ≥ |D| = 2mℓ. Since m = ⌊n/3t⌋ and n is large, we
n
have |H |≥ 2mℓ ≥ 2Cnℓ for C = (1/4t)ℓ.
n
We will use the following result from [22] in our proof of Theorem 2.
Theorem 8. Suppose H is a hereditary L-property. Then the following limit exists.
π(H) = lim |Hn|1/(nr).
n→∞
Moreover, if π(H) > 1, then |Hn|= π(H)(nr)+o(nr), and if π(H) ≤ 1, then |Hn|= 2o(nr).
We now fix some notation. A formula ϕ(x¯;y¯) is trivially partitioned if |y¯| = 0. Given a set X
and n ≥ 1, Xn = {(x ,...,x ) ∈ Xn : i 6= j implies x 6= x }. If M ∈ H, ϕ(x¯;y¯) is a formula,
1 n i j
A ⊆ M|y¯|, and a¯ ,...,a¯ ∈ M|x¯| are pairwise distinct, then define qftpM(a¯ ,...,a¯ ;A) to be the
1 k ϕ 1 k
element p (x¯ )∪...∪p (x¯ ) of S (A) such that M |= p (a¯ )∪...∪p (a¯ ).
1 1 k k ϕ,k 1 1 k k
The following notation is from [3]. Let Index be the set of pairs (R,p) where R(x ,...,x )
1 t
is a relation of L and p is a partition of [t]. Given (R,p) ∈ Index, define R (z¯) be the formula
p
obtained as follows. Suppose p ,...,p are the parts of p, and for each i, m = minp . For each
1 s i i
x ∈ {x ,...,x }, find which part of p contains j, say p , then replace x with x . Relabel the
j 1 t i j mi
variables (x ,...,x ) = (z ,...,z ) and let R (z¯) be the resulting formula. Now let rel(L)
m1 ms 1 s p
consist of all formulas ϕ(u¯;v¯) obtained by permutingand/or partitioning the variables of a formula
of the form R (z¯)∧ z 6= z , where (R,p) ∈ Index.
p V1≤i6=j≤|z¯| i j
Given a formula ϕ(u¯;v¯) and an L-structure M, let ϕ(M) = {a¯¯b ∈ M|u¯|+|v¯| : M |= ϕ(a¯;¯b)}.
Observe that if ϕ(u¯;v¯) ∈ rel(L) and M is an L-structure, then ϕ(M) ⊆ M|u¯|+|v¯|, and |u¯|+|v¯|≤ r.
We will use the fact that any L-structure M is completely determined by knowing ϕ(M) for all
trivially partitioned ϕ ∈ rel(L), or by knowing ϕ(M) for all ϕ(u¯;v¯) ∈ rel(L) with |u¯| = 1. Given
a formula ϕ(u¯;v¯) and n ≥ 1, set
F (n) := {U ⊆ [n]|u¯|+|v¯| : there is M∈ H with ϕ(M) = U}.
ϕ n
We now prove Theorem 3 and then Theorem 2, which we restate here for convenience. Recall H is
a fixed hereditary L-property and the maximum arity of L is r.
Theorem 3. One of the following holds.
(a) VC∗(H)< ∞ and there K > 0 such that for sufficiently large n, |H |≤ nK or
0 n
(b) VC∗(H)= ∞, and there is a constant C > 0 such that for sufficiently large n, |H |≥ 2Cn.
0 n
Proof. If VC∗(H) = ∞, then Theorem 4 implies there is a constant C > 0 such that for large n,
0
|H | ≥2Cn, so (b) holds. Supposenow VC∗(H) = d <∞. Fix ϕ(x¯) a trivially partitioned formula
n 0
from rel(L). Setk = (d+1)r2(r2) and fix n>> k,d,|x¯|. ObserveFϕ(n)⊆ [n]|x¯| because ϕ ∈ rel(L).
We show VC(F (n)) < k. Suppose towards a contradiction VC(F (n)) ≥ k. Then there is
ϕ ϕ
U ⊆ [n]|x¯| of size k shattered by F (n). In other words, for all Y ⊆ U, there is M ∈ H with
ϕ Y n
ϕ(M )= Y. Lemma 1 part (b) implies there is U∗ ⊆ U which is an indiscernible set with respect
Y
to equality, and which has size at least (k/2(|x2¯|))1/|x¯| ≥ (k/2(r2))1/r = d+1. Let V = {v¯1,...,v¯d+1}
consist of d + 1 distinct elements of U∗. Let ρ ∈ S∅ (∅) be such that for all i 6= j, ρ(v¯,v¯ )
2|x¯| i j
holds (this exists because V ⊆ U∗ and U∗ is an indiscernible set with respect to equality). Note
10