Additivity of the dp-rank

The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density.

and appropriate minimality notions. Our simplified notion of rank turned out to be very close to Shelah's κ ict , but localized to a type.
A few characterizations of dp-minimality have since been given in the literature, and similar equivalences can also be applied to higher ranks. It seems clear at this point that part of the strength of this concept is the interaction between those equivalent definitions. We are mainly referring to the "standard" (syntactic) independent array definition as given in [14,8] (see Definition 2.4), which is very useful when one wants to deal with formulas, and a very simple "semantic" variant (see Definition 2.1), which can be found, for the dp-minimal case, in Simon's paper ( [13]), but which (as far as we know) has never been stated for a type with rank greater than 1. In this paper, we will mainly work with the semantic definition, which proves useful and convenient for our purposes. However, throughout the paper we prove quite a few different characterizations of dp-rank, especially in the finite rank case; we summarize them in Theorem 5.4.
As the word "rank" indicates, dp-rank is a certain measure of the "size" of a type. It would probably be more accurate to say that dp-rank measures "diversity" of realizations of the type -how much the realizations differ from each other, as can be seen by external parameters. We will elaborate on this below. Just as with most rank notions, one wonders whether it has basic properties such as subadditivity: the rank of a tuple should be bounded by the sum of the ranks of its elements. This is the main result of Section 4. As a corollary, we obtain global bounds for alternation ranks of formulas in a theory of finite rank (so in particular in a dp-minimal theory).
Dp-rank is a notion which is related to weight in stable theories (and motivated by it), and to certain more recent notions of weight in dependent theories, e.g. [7], [8], and [6]. It is therefore natural to wonder whether dp-rank can play a role similar to weight in dependent theories. Although this is still a line of work which can be pursued (and probably provides natural lines of research), in this paper we give limitations to the analogy, showing that dp-rank fails to have certain properties that one would hope for in a notion of weight, in at least two significant ways.
One of our original intentions (influenced greatly by the weight analogy) was to prove results for dp-minimal types, and then try to extend them to higher ranks by induction, at least for the finite rank case. The analogy with weight in stable (and even simple) theories led us to ask whether every type of finite dp-rank can be "analyzed" to some extent by types of rank 1. This turns out to be not the case, and provides the first limitation to the weight comparison. Example 2.10 presents a theory with types of finite rank but no dp-minimal types. This is a very "tame" strongly dependent theory (hence of "finite weight" -in fact, of dp-rank 2), but there are no types of "weight" 1 (whatever definition of weight one chooses to use; see e.g. [7]).
Another direction where the analogy fails is the finite/infinite correspondence. Hyttinen's results in [4], that yield a decomposition a type of finite weight in a stable theory into types of weight 1, also imply that no type has infinite but "rudimentarily finite" weight. In other words, if a type has weight ℵ 0 , then there exists an infinite set witnessing this. In Subsection 2.2 (particularly, Theorem 2.13) we observe that this fails to be the case for dp-rank, even in stable theories. The example presented there also answers negatively an analogous question concerning Adler's notion of burden in [1] (since burden is essentially the same as dp-rank in dependent theories 1 ). Specifically, this is an example of an ℵ 0 -stable theory (in particular, superstable) such that every type over ∅ has infinite dp-rank. Since in a superstable theory weight of any given complete type is finite, Theorem 2.13 exemplifies a very important difference between weight and dp-rank: weight only takes into account nonforking extensions of the type; so when passing to a forking extension, weight may grow. Dp-rank does not distinguish between different kinds of extensions, so it can only go down when the set of parameters is increased (as is the case with most rank/dimension notions).
In a sense, with the notion of dp-rank, we capture a very particular aspect of the size of the type, and our thesis is that it might be "better" to think of dp-rank in relation to finite combinatorial invariants of Vapnik-Chervonenkis (VC) classes, particularly, VC-density, instead of in terms of weight.
VC-density is one of the possible measures of growth rate of types over finite sets. Specifically, assuming that ∆(x, y) is a finite set of dependent formulas (which is equivalent to the growth rate of the number of finite ∆-types being polynomial rather than exponential), VC-density bounds the degree of the corresponding polynomial. So VC-density 1 corresponds to linear growth of the number of ∆-types over finite sets. We prove in Section 5 that one way of looking at dp-rank is the following: dp-rank of a type p corresponds to the degree of a polynomial that measures how many types can a realization c |= p realize over finite indiscernible sequences. In other words, dp-rank of p is a measure of the number of realizations of p that have different types over finite indiscernible sequences.
So in particular dp-minimality means that realizations of p realize only order of n types over any given indiscernible sequence of length n. This is equivalent, as is shown in Section 2, to the following: subsets of indiscernible sequences that a realization of p can definably pick are simply intervals. These intervals can be, of course, of length 1 (that is, singletons). The important observation here is that a dp-minimal element can not pick (definably) a finite tuple of size bigger than 1 from an arbitrary indiscernible sequence. Or, more precisely, its alternation rank can not be bigger than 2. In this respect the behavior of a dp-minimal element resembles an element in either an o-minimal theory, or the theory of equality.
A number of distinctions between the notions of VC-density and dp-rank should be pointed out. The obvious ones are that we only count number of types over indiscernible sequences (as opposed to arbitrary finite sets), and restrict ourselves to realizations of a given type (the second one is not important -one can look at dp-rank of a partial type as well). Another difference is that, when calculating dp-rank, we do not restrict the set of formulas that one is allowed to use to a finite set. This is why, even in a strongly dependent theory, one can end up with types of dp-rank ω (one obtains more and more types over indiscernible sequences by changing the formulas), one more thing that Theorem 2.13 exemplifies.
However, dp-rank is a very natural model theoretic analogue of VC-density. Part of the strength of model theoretic techniques is the ability to approximate complex phenomena in better behaved structures. Indiscernible sequences have already proved very helpful for such approximations in various contexts. Seeking connections between VC-density and dp-rank is another implementation of this idea.
Since we work with the semantic definition of dp-rank, we need some technical results on indiscernible sequences which are quite interesting on their own. In Section 3 we prove a proposition (Proposition 3.3) which provides a "consistent" way to extend indiscernible sequences in an arbitrary theory. Specifically, in the proofs in Section 4 we are sometimes faced with the following situation: a sequence I is indiscernible over various subsets of a set B, but not (necessarily) over B, and we would like a uniform way of extending I to a longer sequence with the same properties. If the theory is assumed to be dependent (and I is unbounded) one can just take the average type of I over B. However, we are not assuming dependence, and it seems of independent interest to find a general technique allowing this (and more) in an arbitrary theory. Here Shelah's general notion of average type with respect to an ultrafilter comes in handy.
1.1. Structure of the paper. We begin Section 2 with definitions, characterizations, and basic properties. Then we proceed to a few examples, which point out what one can and can not expect from dp-rank.
In Section 3 we develop a consistent way of extending mutually indiscernible sequences, which serves us in Section 4.
In Section 4 we will prove the main result of the paper, the sub-additivity of the dp-rank. We start with the dp-minimal case, then proceed to types of finite rank, and finally the infinite case. There was no need, in fact, to give a separate proof for dp-minimal types: the general finite case can be easily modified to include rank 1 as well. However, the proof for dp-minimal types is much less involved, so we include it in order to exemplify the general principles that are used, and then proceed to the finite case by induction. The proof for infinite ranks is different, but easierinfinite combinatorics is more flexible, and calls for much less precise computations. We conclude the section with some corollaries, such as global bounds on alternation ranks of formulas.
In Section 5 we discuss what is known about the relation between VC-density and dp-rank. Theorem 5.4 summarizes all the main equivalent ways of looking at dp-rank. We also pose a few questions and set up a framework for future work.
Throughout the paper, we will work in a monster model of an arbitrary first order theory T . In particular, although the main object of study is a "dependence rank", at no point do we assume that T is dependent.
We will not distinguish between singletons and finite tuples in the notation. For example, when writing ϕ(x), or "a is in the sort of x", if not specified otherwise, x and a could be tuples.

Dp-rank: definitions and basic properties
We begin with the definition of the main notion investigated in this paper.
Definition 2.1. Let p(x) be a partial (consistent) type over a set A. We define the dp-rank of p(x) as follows.
• The dp-rank of p(x) is always greater or equal than 0. Let µ be a cardinal.
We will say that p(x) has dp-rank ≤ µ (which we write rk-dp(p) ≤ µ) if given any realization a of p and any 1 + µ mutually A-indiscernible sequences, at least one of them is indiscernible over Aa.
• We say that p has dp-rank µ (or rk-dp(p) = µ) if it has dp-rank ≤ µ, but it is not the case that it has dp-rank ≤ λ for any λ < µ. • We call p dp-minimal if it has dp-rank 1.
• We call p dependent if dp-rank of p exists, that is, if it is an ordinal. In this case we write rk-dp(p) < ∞. Otherwise we write rk-dp(p) = ∞. • We call p strongly dependent if rk-dp(p) ≤ ω.
Remark 2.2. It may seem that dp-rank of p depends on the set A; however, we will see in Remark 2.7 below that this is not the case. Hence all the notions in Definition 2.1 above are well defined.
The following is easy and standard. Remark 2.3. The following hold for any (partial) type p(x) and a set A.
(i) p has dp-rank 0 if and only if it is algebraic.
(ii) p is dependent if and only if the rk-dp(p) ≤ |T | + .
(iii) If p is a type over a set A, and p ′ is a type over a set B ⊇ A that extends p, then rk-dp(p ′ ) ≤ rk-dp(p). In particular, if p is dp-minimal, then any extension of it is either dp-minimal or algebraic. Definition 2.1 is a nice semantic characterization of the notion of dp-rank. We find it more convenient for the purposes of this paper than the syntactic definition in [8]. It is also much easier to grasp, in case one is unfamiliar with the concept. However, when working with formulas, it is useful to have a more syntactic notion, and we would like to prove that our "soft" characterization is equivalent to the original one. In case the reader is unwilling to deal with technical concepts, it is possible to skip the following definition and Proposition 2.6 in the first reading. These will not be used almost at all in the main body of the paper (Sections 3 and 4), but they are key for understanding important characterizations of dp-rank in the finite case and the connection to VC-density (Proposition 2.8 and Section 5), as well as the fact that dp-rank of a type does not depend on the set over which it is calculated.
The following definitions were motivated by the original definition of strong dependence by Shelah (see e.g. [12]) and appear in [14] and [7]. Definition 2.4. A randomness pattern of depth κ for a (partial) type p(x) over a set A is an array b α i : i < ω α<κ and formulae ϕ α (x, y α ) for α < κ such that: We will omit A if it is clear from the context.
The following is standard.
Lemma 2.5. Let p be a (partial) type over a set A. Then there is a randomness pattern of depth κ over A if and only if there exists an array b α i : i < ω α<κ and formulae ϕ α (x, y α ) for α < κ that satisfy clauses (ii) and (iii) of Definition 2.4.
Proof. Without loss of generality A = ∅ (note that A contains the domain of p). Given an array satisfying clauses (ii) and (iii) of Definition 2.4, we may assume by compactness that the all the κ sequences b α i are as long as we wish, b α i : i < λ for λ arbitrarily big. Applying Ramsey's Theorem and compactness, one obtains new sequences for which clause (i) of the definition of a randomness pattern holds as well.
One consequence of the Lemma is the following: let p be a type, and let A, B be sets, both containing the domain of p. Then there is a randomness pattern of depth κ for p over A if and only if there exists such a pattern over B.
In [8] we defined dp-rank of a type p(x) as the supremum of all κ such that there is a randomness pattern of depth κ for p(x). The following Proposition shows that Definitions 2.1 above are equivalent to the original ones.
The first appearance of any such equivalence appeared for the dp-minimal case in Lemma 1.4 of [13]. We do not know that anyone has generalized this even for finite dp-ranks (or randomness patterns of finite depth).
Proposition 2.6. The following are equivalent for a complete type p(x) over A. Notice that κ below may be a finite cardinal.
(i) There is a randomness pattern of depth κ for p(x) over A.
(ii) It is not the case that the dp-rank of p(x) is less than or equal to κ. (iii) There exists a set I := {I j } := { a j i i∈I | j ∈ κ} of κ infinite mutually indiscernible sequences over A, and a realization c of p(x) such that for all j there are i 1 , i 2 such that tp(a j i1 /Ac) = tp(a j i2 /Ac).
Proof. The proof of the equivalence of (ii) and (iii) in [13] works exactly for the general case. Also, it is clear that (i) implies (iii).
Assume (iii) and we will prove (i). Let I j := a j i and we will assume that all such I j are indexed by Z, and let ϕ j be the formula such that ϕ j (c, a i1 ) and ¬ϕ j (c, a i2 ) holds. Claim 2.6.1. If {i | ϕ j (c, a j i )} is both coinitial and cofinal, then there is a subsequence I j 0 of I j such that (after re-enumerating the elements) I j 0 := a 0,j i i∈Z and ¬ϕ j (c, a 0,j i ) holds if and only if i = 0.
Proof. The construction of I j 0 is immediate from the definition.
Notice that if in every sequence I j we have that either {i | ϕ j (c, a j i )} or {i | ¬ϕ j (c, a j i )} is coinitial and cofinal, then we can replace I j by the subsequence I j 0 and (replacing ϕ j (x, y) for ¬ϕ j (x, y) if necessary) we would have an instance of (i). Now, if for some j we have that both {i | ϕ j (c, a j i )} and {i | ¬ϕ j (c, a j i )} are not coinitial and not cofinal, then we have that, for example, {i | ϕ j (c, a j i )} is coinitial and {i | ¬ϕ j (c, a j i )} is cofinal. In this case, if we define ϕ j 0 := ϕ j (x, y 1 ) ∧ ¬ϕ j (x, y 2 ) and I j 0 as a sequence a j 2i , a j 2i+1 i∈Z , we would preserve the mutual indiscernibility, we would have instances of both ϕ j 0 (c, a j 2i , a j 2i+1 ) and ¬ϕ j 0 (c, a j 2i , a j 2i+1 ), and {i | ¬ϕ j (c, a j 2i , a j 2i+1 )} would be coinitial and cofinal. Applying the claim to all such sequences we would have an instance witnessing (i).
Remark 2.7. It follows from Lemma 2.5 and the equivalence of (i) and (ii) in Proposition 2.6 that dp-rank of a type does not depend on the set A over which it is computed. In other words, the notion rk-dp(p) in Definition 2.1 is well-defined.
In the finite rank case, we can prove another characterization of dp-rank, in terms of a natural generalization of the notion of alternation rank. Given a formula ϕ(x, y), we define the p-alternation rank of ϕ(x, y) over A as follows: alt Proposition 2.8. The following are equivalent for a partial type p(x) over A and k < ω: There exists a formula ϕ(x, y) and an A-indiscernible sequence I in the sort of y such that for every subset All the indiscernible sequences in the Proposition are presumed to be infinite.
Notice also that what (ii) is essentially saying is that every k-tuple in I is ϕ(c, y)definable for some c |= p.
Proof. (i) =⇒ (ii). We will use the syntactic definition here. If rk-dp(p) ≥ k, then by Proposition 2.6 there is a randomness pattern of depth k for p(x); that is, there are c |= p, formulas ϕ 1 (x, y 1 ), . . . , ϕ k (x, y k ) and A-mutually indiscernible sequences I i = a i j : j < ω in the sort of y i (for i = 1, . . . , k), such that for every j 1 , . . . , j k there is c = c j1,...,j k |= p such that ϕ i (c, a i j ) if and only if j = j i . Let ϕ(x, y 1 . . . y k ) = k i=1 ϕ i (x, y i ), and let I be an A-indiscernible sequence in the sort of y 1 . . . y k defined as follows: I = a j : j < ω , where a j = a 1 j . . . a k j . Choose arbitrary k distinct indices j 1 , . . . , j k . It is easy to see that c = c j1,...,j k is a realization of p such that ϕ(c, a j ) if and only if j ∈ {j 1 , . . . , j k }, as required in (ii). ( Let I be an A-indiscernible sequence and c |= p such that ϕ(c, y) has ≥ 2k alternations in I. We may assume that the number of alternation is finite, and in fact equal to 2k. We prove by induction on k the following claim: Claim. If I is an A-indiscernible sequence of order type Q, and ϕ(c, y) has 2k alternations in I, then there is a randomness pattern (I α , ϕ α ) of depth k for tp(c/A) with I α being segments of I, and ϕ α = ϕ or ϕ α = ¬ϕ for all α.
The base case k = 0 is trivial. So now given I = a q : q ∈ Q and c as in the claim, since the order type of I is Q, there is an infinite initial segment on which ϕ(c, y) is constant; assume for example that ϕ(c, y) holds, and let r ∈ R be the minimal cut such that for any q > r, the truth value of ϕ(c, y) has changed signs twice in the interval (−∞, q). Let q > r be such that ϕ(c, a q ) holds, and there is no sign change between r and q. Let I 0 = a <r ⌢ a (r,q) and ϕ 0 = ϕ. Notice that by indiscernibility and compactness, given any a i ∈ I 0 the type is consistent. By the induction hypothesis, since the order type of I >q = a >q is still Q, and since I >q has at least 2k − 2 alternations of ϕ(c, y), we can find a randomness pattern {I α }, {ϕ α } for c over A, for α = 1, . . . , k − 1 and I α segments of I >q . Now clearly {I α } k α=0 are mutually A-indiscernible (since they are all segments of the same A-indiscernible sequence), and it is easy to see that {I α } α≤k , {ϕ α } α≤k is a randomness pattern for c over A of depth k.
This finishes the proof of the claim, and the theorem.
Assume that in a theory T every type p(x) over ∅ in the sort of x has dp-rank ≤ k. Then for every formula ϕ(x, y), the alternation rank of ϕ(x, y) is bounded by 2k + 1.
Proof. If alt(ϕ(x, y)) ≥ 2k + 2, then there is an indiscernible sequence in the sort of y and some c in the sort of x, that witness this. By (iii) ⇒(i) in Proposition 2.8, rk-dp(tp(c/∅)) ≥ k + 1.
We conclude this section with a a few examples that illustrate things which can not be expected from dp-rank. We will leave some of the technical details of the examples to the reader.
2.1. Theory of dp-rank 2 with no dp-minimal types. As we mentioned above, one might be drawn to think that all the study of theories of finite dp-rank (theories where all dp-ranks are finite) can be reduced to dp-minimal types. This, however is not the case.
Example 2.10. Consider the theory of an infinite set with two dense linear orders < 1 and < 2 , and take the model completion of it. This is, the theory of a structure M in L := {< 1 , < 2 } such that any finite formula consistent with < 1 and < 2 being dense linear orders is realized in M .
Every one type has dp-rank 2, and there are no dp-rank 1 types: First of all, the model completion exists and by definition it is complete and has elimination of quantifiers in the language L := {< 1 , < 2 }, so tp(a/A) can be understood by formulas of the form x < 1 a, x < 2 a, and x = a for suitable choices of a ∈ A.
It is not hard to show now that given any set A, any 1-variable type p(x) ∈ S(A) has dp-rank 2. Given any such set and type, it is enough to find mutually Aindiscernible sequences a i and b j such that for every k, ℓ we have: • The set is consistent. For this, it is enough to find for every m < ω, sequences a i and b j for i, j < m such that for every k, ℓ < m we have: Such a i and b j can be found by the definition of a model companion. This implies that every type in this theory has dp-rank at least 2, and in particular that there are no dp-minimal types. On the other hand, it is easy to see that no one-type (over any set) has dp-rank bigger than 2.
2.2. Type of dp-rank ω in a theory with types of finite weight. People have asked whether or not strong dependence was equivalent to every type having finite dp-rank, in the same way that a stable theory is strongly dependent if and only if every type has finite weight (some people asked this for Adler's notion of burden [1], which is essentially the same question). Specifically, the question is whether or not it is possible to have randomness patterns of arbitrarily large finite depths but no randomness pattern of infinite depth for a given complete type p(x) (once one forgets the type and just asks whether there is a strongly dependent theory with arbitrarily deep randomness patterns, the question becomes much easier).
The following provides an example that the above is possible even in stable theories. Let and let < S (which we sometimes denote simply by < for simplicity) be the partial order on S defined by (m 1 , n 1 ) < S (m 2 , n 2 ) if and only if n 1 < n 2 . If s = (m, n), we say that s is of level n, and write lev(s) = n. Note that s ≤ t if and only if s is of a smaller level than t (or s = t). So s = t are incomparable if and only if they are of the same level. Hence there are finitely many t's which are incomparable to a given s -in fact, the number is exactly the lev(s).
So we have: Observation 2.11. The following hold for (S, ≤ S ).
(i) For any s ∈ S there are finitely many s ′ ∈ S which are not greater than s.
(ii) For any n ∈ N there exists s 1 , . . . , s n ∈ S such that s i is incomparable to Now consider the theory T ∀ in the language L := {E s } s∈S which states that all E s are equivalence relations and for any s ≤ S t we have ∀x, y, xE s y ⇒ xE t y.
Let T be the model completion of T ∀ , so in particular it is a complete theory with elimination of quantifiers.
T is axiomatized by the following axioms. In order to make things more uniform, let us refer to equality as the unique s of level -1.
• Every class of every E s for s ∈ S is infinite.
• Every E s has infinitely many classes.
• Whenever −1 ≤ n < m, lev(s) = n, lev(t 1 ) = lev(t 2 ) = . . . = lev(t k ) = m, A i is an equivalence class of E ti (for i = 1, . . . , k), then ∩ k i=1 A i contains infinitely many classes of E s . Even without proving the existence of the model completion, one can show directly, given the above axioms and using a standard back and forth argument, that T is a complete theory with elimination of quantifiers.
Proof. For every (non-generic) type p over a countable model M , let lev(p) be the least n, such that there exists a ∈ M and s of level n, such that the formula xE s a is in p. Since this determines which E t classes p "chooses" for s < t, there are only finitely many equivalence relations left to "settle" -specifically, all those E t for which s < t (see Observation 2.11 above). It follows (by quantifier elimination) that there are countably many types of each given level n. Since there are countably many levels, we are done.
We can now show that T is an example of a theory with non finite dp-rank and no randomness pattern of depth ω for the unique type p(x) over the empty set.
Theorem 2.13. For any element c and any natural number n, one can find I such that I is a set of n mutually indiscernible sequences (over ∅) none of which is indiscernible over c. However, no such example can be found with I infinite.
Proof. Let n be a natural number, and let s 1 , . . . , s n be incomparable elements of S of level n (see Observation 2.11). By the axioms of T we can find I 1 , . . . , I n mutually indiscernible sequences of singletons with the following properties: • Elements of I i are E sj -equivalent if and only if j = i, • c is E si -equivalent to the first element of I i for all i.
Clearly, none of the I i 's is indiscernible over c, as required. On the other hand, by Claim 2.12, T is superstable, hence strongly dependent (this is very easy to see directly, but see [12]). So there is no infinite randomness pattern for any type in any model of T . (Alternatively, using quantifier elimination, one can easily give a direct proof that there is no infinite randomness pattern.) Remark 2.14. In a similar fashion, one constructs an example of a theory which is not superstable, but still strongly dependent, with a type of dp-rank ω. This is done by switching the "nesting order" of the equivalence relation in the example above; that is, we demand in the universal theory that for s ≤ S t we have ∀x, y, xE t y ⇒ xE s y.
Here one has to give a direct argument as to why there is no infinite randomness pattern; but this quite straightforward, using quantifier elimination, and we leave it to the reader.
Recall that in a strongly stable theory (strongly dependent and stable) every type has finite weight ( [1,14]). However, in both examples discussed above, the unique (non-algebraic) type over ∅ has dp-rank ω. So we have two examples of theories where every type has finite weight but there are no non algebraic types of finite dp-rank over ∅. In fact, the generic type over any set will also have dp-rank ω by a similar argument. In the example discussed in Remark 2.14 the situation is even more extreme, because almost all types have infinite dp-rank (besides the algebraic and the strongly minimal ones).

Extending indiscernible sequences
Since the main definition of the paper involves mutually indiscernible sequences, it would be nice to have certain tools for handling such "independent arrays". Specifically, we would like to have a "consistent" way of extending indiscernible sequences. Our technique will make use of Shelah's notion of average types with respect to ultrafilters, which provides a way of constructing co-heir extensions. In spite of its usefulness, this notion does not yet seem to be widespread in the model theoretic community.
First, we recall the following easy observation. This fact provides a natural and standard way to extend indiscernible sequences. However, it will not always be good enough for us, since, as mentioned in the introduction, we will need to extend sequences preserving indiscernibility over different subsets.
This is why the definition of average types from Shelah becomes useful: Definition 3.2. Let I = a i be an indiscernible sequence and U an ultrafilter on the index set of I. Given any set B we will define Avg U (I, B), the average type of I over B given by U, as the unique complete type p(x) such that for every formula ϕ(x, y) and b ∈ B we have This definition will allow us to prove the following. The proof requires some knowledge of ultrafilters. Proposition 3.3. Let I be an indiscernible sequence indexed by an order with no last element, and let B be any set. Then, for any indexing set λ, there is an indiscernible sequence I * indexed by λ such that I ⌢ I * is indiscernible over A whenever A ⊆ B is a set such that I was already indiscernible over A.
Proof. We will prove the result for any finite I * . The general case follows by compactness.
If we take U to be an ultrafilter over I such that every set in U is unbounded in I, it follows from the definition that for any A such that I is indiscernible over A and any a 0 |= Avg U (I, AI) we have I ⌢ a is indiscernible over A. Since the definitions imply that Avg U (I, BI) extends Avg U (I, AI) for any such A ⊂ B, any realization a 0 of the type p(x) := Avg U (I, BI) will be such that I ⌢ a 0 is indiscernible over A. Inductively, it follows from the properties of average types that if we let a n+1 |= Avg U (I, BI ⌢ a n , a n−1 , . . . , a 0 ) then I ⌢ a n+1 , a n , a n−1 , . . . , a 0 is an indiscernible sequence over A. This construction will therefore give us an indsicernible sequence I ′ , indexed by n, such that I ⌢ I * is indiscernible over any A over which I was indiscernible (for an arbitrarily large n); by compactness, given any indexing set λ, we can find an indiscernible sequence I * indexed by λ such that I ⌢ I * satisfies the conclusion of the proposition. Proposition 3.3 is the only instance where we will use average types. If unwilling to think about ultrafilters, the reader can just assume the existence of a way to extend indiscernible sequences given by Proposition 3.3. Then I ⌢ I * and J are mutually indiscernible over A.
Proof. It follows from Proposition 3.3 that I ⌢ I * is indiscernible over AJ. Now, if J was not indiscernible over I ⌢ I * there would be a finite tupleā ⌢b withā ∈ I and b ∈ I * such that J is not indiscernible over Aāb. Since I ⌢ I * was indiscernible over AJ, and since I is infinite, we know that there are someā ′ ,b ′ ∈ I such that tp(ā ′b′ /AJ) = tp(āb/AJ).
But this would imply that J is not indiscernible over AI, contradicting our hypothesis.

additivity of the dp-rank
In this section we will prove the (sub-)additivity of the dp-rank (Theorem 4.8), which is the main result of this paper. 4.1. Warm up case: dp-minimal. The first technical lemma essentially deals with sub-additivity for dp-minimal types. It will also form the induction base for the general case. Although we could modify the proof of the general statement slightly so that it deals with rank 1 as well, we decided to include the simple base case explicitly, since it exemplifies the general technique that we are using. Proof. We will do an induction on n. Since any extension of a dp-minimal type is dp-minimal (or algebraic), if n = 1 there is nothing to prove.
Assume now that I := {I 1 , . . . , I n+1 } is a set of mutually B-indiscernible sequences for B ⊃ A. By definition {I 1 , . . . , I n } are mutually indiscernible over BI n+1 so we can, by the induction hypothesis, find n − 1 of the I j 's which are mutually indiscernible over BI n+1 a; we may assume without loss of generality that {I 1 , . . . , I n−1 } are mutually indiscernible over BI n+1 a. If I n+1 was indiscernible over {a} ∪ B ∪ {I 1 , . . . , I n−1 } the sequence {I 1 , . . . , I n−1 , I n+1 } would satisfy the conditions of the claim, so we may assume that this is not the case. Since non indiscernibility can be witnessed by a finite sequence, we will assume for the rest of the proof that I n+1 is not indiscernible over Bab for someb ∈ {I 1 , . . . , I n−1 }, and that {I 1 , . . . , I n−1 } are mutually indiscernible over I n+1 Ba. Proof. For each k with 1 ≤ k < n we will inductively define a "continuation" I * k of I k in the following way: Suppose we have picked I * j for j < k, and define I * k be a sequence indexed by ω as in Proposition 3.3 for I = I k and B = B ∪ n i=1 I i ∪ k−1 j=1 I * j ∪ {a} so that given It follows from the construction and Corollary 3.4 that • {I 1 ⌢ I * 1 , . . . , I n−1 ⌢ I * n−1 , I n , I n+1 } is a set of mutually B-indiscernible sequences, • {I 1 ⌢ I * 1 , . . . , I n−1 ⌢ I * n−1 } is mutually indiscernible over I n+1 Ba, and • I n+1 is not indiscernible over Bab for someb ∈ {I 1 , . . . , I n−1 }.
Since the sequences in {I 1 ⌢ I * 1 , . . . , I n−1 ⌢ I * n−1 } are mutually indiscernible over I n+1 Ba, there is an automorphism fixing I n+1 Ba and sendingb to someb ′ ∈ {I * 1 , . . . , I * n−1 }. Now we have • {I 1 , . . . , I n−1 , I n , I n+1 } is a set of Bb ′ -mutually indiscernible sequences, • {I 1 , . . . , I n−1 } is mutually indiscernible over I n+1 Bb ′ a, and • I n+1 is not indiscernible over Bb ′ a, which, replacing B with Bb ′ , is precisely the conditions we started with plus the conclusion of the claim. Since any n-subset of mutually Bb ′ a-indiscernible sequences of {I 1 , . . . , I n−1 , I n , I n+1 } would in particular be Ba-indiscernible, the claim is proved. Now the lemma follows almost immediately. Since {I 2 , I 3 . . . , I n , I n+1 } are mutually indiscernible over I 1 B, there must, by induction hypothesis, be a subset of n−1 mutually I 1 Ba-indiscernible sequences. But such set cannot contain I n+1 since, by hypothesis given in Claim 4.1.1, this sequence is not (by itself) indiscernible over Ba. So {I 2 , I 3 . . . , I n } are mutually indiscernible over I 1 Ba. In exactly the same way we can prove that {I 1 , I 3 . . . , I n } are mutually indiscernible over I 2 Ba which in particular implies that I 1 is indiscernible over B ∪ {I 2 , I 3 . . . , I n } ∪ {a}. So {I 1 , I 2 , I 3 . . . , I n } are mutually indiscernible over Ba, as required.
Corollary 4.2. (Sub-additivity of dp-rank for dp-minimal types) Let tp(a i /A) be dp-minimal for 1 ≤ i ≤ k. Then the dp-rank of tp(a 1 . . . a k /A) is at most k.
Proof. By induction on k. For k = 1 there is nothing to do. Assume that the Corollary holds for all sets B and tuples of less than k dp-minimal (over B) elements. Now fix A and a 1 , . . . , a k dp-minimal over A. Let I 1 , . . . , I k , I k+1 be mutually indiscernible over A. By the lemma above, without loss of generality, I 1 , . . . , I k are mutually indiscernible over Aa k , call it B.
Recall that extensions of dp-minimal types have rank at most 1, so we may assume that a 1 , . . . , a k−1 are dp-minimal over B. Hence by the induction hypothesis, dp-rank of the tuple a 1 . . . a k−1 over B is at most k − 1. By definition, one of the sequences I 1 , . . . , I k is indiscernible over Ba 1 . . . a k−1 = Aa 1 . . . a k , which is exactly what we needed. Remark 4.3. Notice that in Claim 4.1.1 we did not assume that I k is indiscernible over I n ∪ {a}. That is, we know that I k (for k < n) is indiscernible over BI =k and over BI =k,n a, but not necessarily BI =k a. This (and the analogue issue in Lemma 4.6) is the reason we could not work using just Fact 3.1, and decided to use the extensions described by Proposition 3.3.

4.2.
The finite case. The following proposition, from which the main result of this section will follow easily, is a generalization of Lemma 4.1. To prove Proposition 4.4, we rephrase the statement in a way that will allow us to do an easy induction. For this we will need to following definition. So in particular, with this notation, a type p(x) over A has dp-rank less than or equal to k if and only if for any realization a of p(x) and every set I of mutually indiscernible sequences where |I| > k, we have that I, a satisfies S k,1 .
With this notation we can state a generalization of Proposition 4.4, the proof of which will admit a clear induction argument. We will start by proving the following analogue of Claim 4.1.1.
Lemma 4.6. Let a be an element, and let I be a set of mutually A-indiscernible sequences. Let J be a subset of I and I ∈ I be such that J is mutually indiscernible over AIa and such that I is not indiscernible over AJ a. Then we can extend A to a set B such that the following hold: • I is mutually indiscernible over B.
• I is not indiscernible over Ba.
• J is mutually indiscernible over Ba.
Proof. We will assume that J is finite, which is the case we need for Theorem 4.8. However, the general case follows exactly in the same manner, using ordinal enumerations of the sequences in J and transfinite induction. We can enumerate J := {J 1 , . . . , J n } and, as in the proof of Claim 4.1.1, define a "continuation" J * t := a * i i∈ω for every sequence J t ∈ J inductively (on t) having that J ⌢ t J * t is indiscernible over any  Proof. Notice that we have already proved the result assuming k = 1. It is enough to show that S k,n implies S k,n+1 , and we will show this by induction on n (for a fixed k). So let I be a set of m mutually B-indiscernible sequences, a be an element such that I, a satisfies S k,i for all 1 ≤ i ≤ n (so in particular, it satisfies S k,n and S k,1 ) and let I ′ := {I 1 , . . . I k+n+1 } be a subset of I; we will prove that I ′ contains a subset of size n + 1 of sequences which are mutually indiscernible over Ba.
Let I i be any sequence in I ′ . Since I ′ \{I i } is a set of n+k mutually indiscernible sequences over BI i , there is a subset I i of size n which are mutually indiscernible over BI i a. If I i is indiscernible over BI i a then we would have a set of size n + 1 of mutually indiscernible sequences over Ba and the proposition would be satisfied. So we may assume towards a contradiction that for every i the sequence I i is not indiscernible over BI i a.
Now, for each i we apply Lemma 4.6 extending B until we get I i not indiscernible over Ba for all i and I ′ are mutually indiscernible over B. This contradicts S k,1 of I (and S k,n too).
This completes the proof of Proposition 4.4.
Proof. Let I := {I 1 , . . . , I k1+k2+1 } be mutually A-indiscernible sequences. By Proposition 4.4 applied to a 1 , I, there is a subset I 1 of I of size k 2 + 1 of sequences which are mutually indiscernible over Aa 1 . By definition of dp-rank of tp(a 2 /Aa 1 ), we get that there is a sequence I ′ ∈ I 1 which is indiscernible over Aa 1 a 2 . By definition of dp-rank, this completes the proof of the theorem.
We get the following corollary (compare with Corollary 4.12).

Corollary 4.9. Let T be any theory.
If all the one variable types have finite dp-rank, then every type (with finitely many variables) in the theory has finite dp-rank.
If all the one variable types have dp-rank ≤ k, then every type (with finitely many variables) p(x) has dp-rank ≤ |x| · k.
The following follows immediately from Theorem 4.8 and Proposition 2.8. Corollary 4.10. Let T be any theory, and assume that all the one variable types have dp-rank ≤ k. Then for every formula ϕ(x, y) we have alt(ϕ(x, y)) ≤ 2k|x| + 1.

4.3.
The infinite case. The proof of sub-additivity for the infinite case is in fact much easier than in the finite case.
Theorem 4.11. Let I be a set of κ mutually indiscernible sequences over A and a an element such that S κ,1 holds for I, a. Then S κ,κ holds for I, a. In particular for any cardinal numbers κ, λ and any tuples a, b we have that rk-dp(ab/A) ≤ max(κ, λ), whenever rk-dp(a/A) ≤ κ and rk-dp(b/A) ≤ λ.
Proof. Let I, a be any pair satisfying S κ,1 . We can partition I = α∈κ I α into a disjoint union of κ many sets of κ many sequences. By hypothesis we know that for any β the set I β are mutually indiscernible over A ∪ α =β I α so by assumption we have that some sequence I β in I β is indiscernible over A ∪ α =β I α ∪ {a}. Doing this for any β ∈ κ we get a set of sequences {I α } α∈κ such that I β is indiscernible over A ∪ α =β I α ∪ {a} which by definition proves that I, a satisfies S κ,κ . The rest of the proof follows exactly as in Theorem 4.8.
Since this immediately implies that rk-dp(ab/A) ≤ ω whenever rk-dp(a/A) ≤ ω and rk-dp(b/A) ≤ ω, this theorem provides a very easy proof of the fact that strong dependence (rk-dp(p(x)) ≤ ω for all p(x) or, equivalently, no randomness pattern of depth ω) only needs to be verified in one variable. Summarizing, we get the following (very) easy corollary, which was originally proved by Shelah in [12] (Observation 1.6). Also, we get a dp-rank version of Shelah's theorem that T is dependent if and only if the independence property cannot be witnessed by ϕ(x, y) with |x| = 1.
Recall that a theory is dependent if and only if every type is dependent, that is, rk-dp(p(x)) ≤ |T | + for any type p(x) 2 . So the following, which follows immediately from Theorem 4.11, is a new (and simpler) proof of Shelah's Theorem II.4.11 in [9].

VC-density
Recent results by Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko [2], show that in many of the well behaved dependent theories, the V C-density can be calculated and, in many of the cases that they considered, it is linear. In this section we define V C-density (or rather, a dual notion, which we call V C * -density 3 ), and discuss some connections between it and dp-rank.
For the sake of clarity, we introduce a few notations. Let p(x) be a type, and B a set; we denote the set of all complete types over B which are consistent with p(x) by S p(x) (B). If ϕ(x, y) is a formula, or ∆(x, y) is a set of formulas, we can speak of 2 We are defining a type to be dependent depending of the behavior of the "dual" formula.
Reasons why this is the right notion can be found in Observation 2.7 in [7]. 3 Given a family ∆(x, y), the V C * ∆ -density will correspond to the V C-density of the family ∆ after we invert the roles of x and y.
ϕ(x, y)-types, or ∆(x, y)-types (over B) consistent with p(x); these will be denoted by S p(x) ϕ and S p(x) ∆ , respectively. We also remind the reader of some basic notation for asymptotic behavior of functions. Let f, g : N → R + . We write that f = O(g) if for some constant r > 0 for any n large enough we have f (n) ≤ r · g(n).
Recall that if f is a polynomial function, its order of magnitude is completely determined by its degree. That is, if f, g are polynomials, then f = O(g) if and only if deg(f ) ≤ deg(g).
Definition 5.1. Let C be a large, saturated enough, model of T , let ∆(x, y) be a finite set of formulas in the language of T , and let p(y) be a (partial) type over a set of parameters in C. The V C * -dimension of p(y) with respect to ∆ is greater than or equal to n if there is a set A of size n such that, for any A 0 ⊂ A there is some b |= p(y) and some δ(x, y) ∈ ∆ such that for any a ′ ∈ A, we have Whenever this happens we will say that ∆ shatters A with realizations of p(y).
We will say that the V C * -dimension of p(y) with respect to ∆ is n if the V C *dimension is greater than or equal to n, but not greater than or equal to n + 1.
Notice that if ∆ shatters a set A with respect to p(y) ∈ S k (B), then every subset of A is (externally) definable as where b varies among realizations of p(y) (and δ ∈ ∆). If ∆ is a singleton (this is, if there is a single formula δ(x, y) in ∆) this is of course equivalent to saying that p(y) is consistent with 2 |A| different ∆-types over A. So, if instead of counting subsets we count types, we will get a notion that, although it is not exactly the same as V C * -dimension when ∆ is not a singleton, it is closely related to this notion (particularly asymptotically). Recall that by S p(y) ∆ (A) we denote the set of all ∆-types over A consistent with p(y). With this notation, we can look for the largest n such that there is some set A of size n such that |S p(y) We are slowly getting to the notion of V C-density that we will work with. It was proved (apparently independently by Sauer, Shelah, and Vapnik-Chervonenkis) that if the V C * -dimension of ∆ with respect to p(y) is equal to d, then, for any A of size greater than d, the number π (∆,p) (A) of subsets of A externally definable with realizations of p, satisfies π ∆,p (A) ≤ |A| 0 + · · · + |A| d .

Now, |S
p(y) ∆ (A)| is bounded by |∆| times π ∆,p (A). This implies that when the V C * -dimension of ∆ with respect to p is d, then each formula in ∆ can define (using parameters in p), at most O(|A| d ) externally definable sets of A. It follows that if we vary A among increasing subsets of an infinite set, we get polynomial growth of the number of ∆-types |S p(y) ∆ (A)| over A, and a very natural question to ask is whether d is the best bound on the degree of the polynomial. This prompts the following definition of V C * -density of a type. We will define (adapting the notions in [2]) the V C * ∆ -density of a type p(y) over a set C to be inf{r ∈ R ≥0 | |S p(y) What we formally mean by this is that there exists a function f : N → R + such that f = O(n r ), and | S p(y) If in the definition above A is allowed to range over all finite sets (that is, C = M for some M |= T saturated enough), we omit "over C", and simply say "V C * ∆ -density of p".
Using our present notation, Proposition 2.8 implies that if rk-dp(p(y)) ≥ k, then there exists a formula ϕ(x, y) and an indiscernible sequence I in the sort of y, such that |S p(y) ϕ (I ′ )| ≥ n k for every I ′ ⊆ I of size n ≥ k. This of course means that V C * ϕ -density of p(y) is at least k. In order to make this connection between dp-rank and V C * -density more precise, we state the following proposition.
Recall (Proposition 2.6) that rk-dp(p) ≥ k if and only if there is a randomness pattern I α , ϕ α of depth k for p. Below we will say that rk-dp(p) ≥ k is witnessed by formulas in ∆ (where ∆ is a set of formulas) if all ϕ α are in ∆.
Proposition 5.2. Let p(y) be a type over A and ∆ be a set of formulas which is closed under boolean combinations. Then the following are equivalent.
(ii) There is an A-indiscernible sequence I and some formula ϕ(x, y) ∈ ∆ such that p(x) has V C * ϕ -density at least k over I. (iii) There is an A-indiscernible sequence I and some formula ϕ(x, y) ∈ ∆ such that p(x) has V C * -density bigger than k − 1 with respect to ϕ(x, y) over I.
Proof. (i) =⇒ (ii) by Proposition 2.8, as explained above (note that the formula one gets in Proposition 2.8 is a boolean combination of ∆-formulas, hence is itself in ∆), and (ii) =⇒ (iii) is trivial. (iii ) =⇒ (i). Let f : N → N be the following function: f (n) = |S p(y) ϕ (I ′ )| for some/every I ′ ⊆ I of size n. It follows from the assumption that f is not O(n k−1 ). This means that for every r > 0 there is n such that f (n) > r · n k−1 .
We may assume that the order type of I is R (indeed, all we need is to keep |S p(y) ϕ (I ′ )| for all I ′ ⊆ I finite). We write I = a r : r ∈ R . In order to continue, we will need the following definition. Given some c |= p, an element r ∈ R will be defined to be a "switch point in I for c" if there is some ǫ ∈ R such that either ϕ(c, a r−δ ) ⇔ ¬ϕ(c, a r ) for all δ < ǫ, or ϕ(c, a r+δ ) ⇔ ¬ϕ(c, a r ) for all δ < ǫ. Proof of the claim: Assume this is not the case. Let I ′ ⊆ I finite, denote n =: |I ′ |. Let q ∈ S p(y) ϕ (I ′ ), c |= q. Below we refer to a cut of I ′ (by which we mean either an element of I ′ or an interval between two adjacent elements in I ′ ) as a "switch cut for c in I ′ ", if the interval it induces in I contains a switch point of c in I. Note that there are 2n + 1 cuts in I ′ , hence at most 2n+1 k−1 choices for possible sequences of switch cuts (by the assumption towards contradiction).
Notice that q is completely determined by knowing (1) the switch cuts for c in I ′ , (2) what are the signs of ϕ(c, y) of different segments between the switch cuts of c in I ′ , and (3) the sings of the cuts themselves. Since there are at most 2n+1 k−1 choices for possible sequences of switch cuts, at most 2 k possible sequences of signs on the segments between the switch cuts, and at most 2 k−1 possible signs for the switch cuts, we would get that | S p(y) ϕ (I ′ )| = O(|I ′ | k−1 ) for all I ′ , contrary to the assumption.
Now, let J be the sequences of pairs of I J = a q2i , a q2i+1 : i < 2k , and let ψ(x 1 , x 2 ; y) be the formula ϕ(x 1 , y) ↔ ϕ(x 2 , y). Clearly, J is an A-indiscernible sequences and since ∆ is closed under boolean combinations, ψ(x 1 , x 2 , y) ∈ ∆. Finally, by construction alt p(y) A (ψ) is at least 2k witnessed by J, so by Proposition 2.8, we have rk-dp(p) ≥ k, witnessed by ψ ∈ ∆, as required.
Remark 5.3. Notice that we needed to define switch points, of which at first there may seem to be as many as the alternation rank. But there is the subtle issue that "isolated points" only count as one switch point, even though they contribute to two for the alternation rank. In fact, by changing the sequence and the formula, we manage to ensure that all those "switch points" happen on "isolated" points (now pairs), each of which then contributes two alternations, hence obtaining alternation rank 2k.
Proposition 5.2 explains why in the example of non-integer V C-density presented in [2] one has to work over sets that are not indiscernible, and why over indiscernible sequences V C-density becomes an integer: in this case, V C-density simply equals the appropriate dp-rank.
We now combine Propositions 2.6, 2.8, and 5.2, and summarize all the main characterizations of finite dp-rank that we have shown in this article.
Theorem 5.4. The following are equivalent for a type p(y) over a set A: (i) rk-dp(p) ≥ k.
(ii) There is a randomness pattern of depth k for p(x) over A.
(iii) There is a formula ϕ(x, y) and an A-indiscernible sequence I in the sort of x such that the V C * ϕ -density of p over I is at least k. (iv) There is a formula ϕ(x, y) and an A-indiscernible sequence I in the sort of x such that the V C * ϕ -density of p over I is bigger than k − 1.
(v) There is a formula ϕ(x, y) and an A-indiscernible sequence I in the sort of x such that for every I ′ ⊆ I of size k there exists c |= p satisfying ϕ(x, c) ∩ I = I ′ (that is, every subset of I of size k is externally ϕ(x, y)definable by a realization of p). (vi) There is a formula ϕ(x, y) with alt p(y) One may obtain a more precise (but also more technical) version of the theorem by restricting to a set of formulas ∆ closed under boolean combinations, as in Proposition 5.2.
Remark 5.5. In the proof of (iii) =⇒ (i) in Proposition 5.2 we only needed that |S p(y) ϕ (I ′ )| is not O(|I ′ | k−1 ), whereas the assumption gives more: not O(|I ′ | s ) for some s > k − 1. Along this line, we note that using clause (v) in Theorem 5.4, one can deduce that the following statements are also equivalent to rk-dp(p) ≥ k. The statements are more technical, but the equivalences are stronger.
(3) There is ϕ(x, y) and I such that |S p(y) ϕ (I ′ )| is not O(|I ′ | k−1 ) for I ′ finite. Where f = Ω(g) means g = O(f ), whereas f = ω(g) means that f strictly dominates g up to any multiplicative constant, that is, for every constant r > 0 we have f (n) > r · g(n) for all n large enough.
Note that in order to go from (3) to (1) one may need to change the formula and the indiscernible sequence (just like in the equivalence of (iv) and (v) in the theorem), and this is crucial. One may ask whether similar statements hold with the same formula and sequence. We have not given it much thought.
We have recently learned that Vincent Guingona and Cameron Hill have investigated V C * -density (and other properties) over indiscernible sequences in much greater detail in [3].
Notice that although Proposition 5.2 demonstrates a nice connection between dprank and V C * -density, it is still quite unsatisfactory. One would hope to connect V C * -density in general to dp-rank. For example, all known example of dp-minimal theories seem to have V C * -density 1 (most of what is known has been proved in [2]). Is this a coincidence, or an example of a deep connection? Specifically, we ask: Question 5.6. Does every dp-minimal theory have V C * -density 1?
It is not so clear how to approach the general question. The proofs in [2] are very case-specific and difficult. Any statement which states a bound for the V C *density in terms of the dp-rank, would need to involve achieving finite indiscernible sequences, hence require nontrivial combinatorial arguments. Some partial results have been obtained by the authors in a subsequent work, but not much is known in general.
Thinking about the possible arguments, it came to our attention that things could be much more manageable if we could concentrate in single variables; by this we mean that both definitions -of dp-rank and V C * -density-could be made by looking at the behavior of the realizations of the type with respect to singletons (for precise statements, see the two questions that follow this discussion). This sort of result is not uncommon in model theory: A theory is dependent if arbitrarily large sets of elements (not tuples) can not be shattered; if a dependent theory is unstable then the strict order property can be witnessed with elements, etc. So it would not be too surprising if both V C * -density and dp-rank could be defined by just looking at the singletons. The following question appeared in a first version of this paper.
Question 5.7. If p(x) is a (partial) type over A of dp-rank greater than n, can this be witnessed by indiscernible sequences of elements? This is, are there I 1 , . . . , I n mutually A-indiscernible sequences of singletons and some c |= p(x) such that I j is not indiscernible over Ac for all 1 ≤ j ≤ n?
This was proved to be false: there are theories which are not dp-minimal but such that given any element and any two mutually indiscernible sequences of singletons (over the empty set), at least one of them is indiscernible over the element. However, since the theory is not dp-minimal, you can find a type over the empty set with dp-rank bigger than 1, thus providing a counterexample to the question even with A = ∅.
The question, however, turned out to be the wrong question. The following was proved by Kaplan and Simon in [5]: Fact 5.8. [5] If p(x) is a (partial) type over A of dp-rank greater than n, there is an extension q of p over some B ⊃ A such that q has dp-rank greater than n, witnessed by indiscernible sequences of singletons.
The second questions is concerned with the behavior of V C * -density: Question 5.9. Suppose that p(y) is a type such that for all ∆(x, y) := {δ 1 (x, y), δ 2 (x, y), . . . , δ n (x, y)} where x is a single variable, we have that the V C * -density of p(y) with respect to ∆ is greater than d. Is d the V C * -density of p(y) with respect to any ∆?
Notice that a positive answer to Question 5.9 would imply that we could define the VC-density of a type by considering formulas ∆ for whichx is a singleton. If this were true, we would have more tools and evidence for establishing a tighter connection between dp-rank and V C * -density.