aa r X i v : . [ m a t h . L O ] J u l ON VC-DENSITY OVER INDISCERNIBLESEQUENCES
VINCENT GUINGONA,CAMERON DONNAY HILL
Abstract.
In this paper, we study VC-density over indiscerniblesequences (denoted VC ind -density). We answer an open questionin [1], showing that VC ind -density is always integer valued. Wealso show that VC ind -density and dp-rank coincide in the naturalway. Introduction
In recent years, the examination of NIP (or dependent) theories hasbeen especially fruitful. NIP includes many theories of general mathe-matical interest, such as o-minimal theories (including the theory of thereal numbers) and C-minimal theories (including algebraically-closedvalued fields). While there have been several remarkable successes, noone has yet provided a decisively “correct” notion of super-dependencein analogy with super-stability. However, one viable candidate notionis that of strong-dependence addressed by Shelah in [13], [14], [15] and[16]. Associated with the notion of strong-dependence is the notion ofdp-rank, one of the subjects of this article. In some sense, dp-rank isindeed a measure of the complexity of families of definable sets relativeto a theory. In a recent paper of Kaplan, Onshuus, and Usvyatsov [8],they show that dp-rank is subadditive in the way one would expect adimension to be (see Theorem 3.4 below, which is Theorem 4.8 of [8]).Following a somewhat different line of investigation, Aschenbrenner,Dolich, Haskell, MacPherson, and Starchenko in [1] suggest using VC-density as a dimension for strongly-dependent theories. In fact, thethesis of [1] holds that VC-density is, in some sense, a more “accurate”measure of the complexity of a formula than the competitors, possiblyincluding dp-rank. We say “possibly,” for the relationship betweendp-rank and VC-density is not well understood. (It is known that VC-density bounds dp-rank from above – see Proposition 1.4 below – but itis not clear that VC-density also provides a lower bound for dp-rank.)
Date : November 18, 2018.2010 Mathematics Subject Classification. Primary: 03C45.
This paper amounts to a first attempt to establish that relationship.In order to make headway, we simplify the situation by analyzing VC ind -density – VC-density evaluated exclusively relative to indiscernible se-quences – instead of the full VC-density. In this restricted context,we answer several of the open questions around the relation betweendp-rank and VC-density. Along the way, we establish some additionalinteresting results; for example, we show that VC ind -density is alwaysinteger-valued (Theorem 1.6 below).The structure of this article is as follows. In the remainder of thissection, we provide most of the important definitions and state themain results. In Section 2, we introduce a notion of “local” dp-rank,and we show that it is, indeed, the “correct” localization of the standarddp-rank. Though its theory is not deep in itself (or as relates to dp-rank), the local dp-rank is rather interesting in its relation to VC ind -density. Indeed, in Section 3, we show that local dp-rank and VC ind -density coincide. We derive Theorems 1.6 and 1.7 as straightforwardcorollaries. Finally, in Section 4, we briefly discuss how the techniquesemployed in this paper to understand VC ind -density might be appliedto the original questions surrounding VC-density.1.1.
VC-density and VC ind -density.
Let T be a complete first-order theory in some language L and let C be a monster model for T .Let p ( x ) be a partial type and let ϕ ( x ; y ) be any formula. For any set B ⊆ C y , let S ϕ ( B ) ∩ [ p ] denote the set of all ϕ -types over B consistentwith p ( x ). That is, all maximal subsets of { ϕ ( x ; b ) : b ∈ B } ∪ {¬ ϕ ( x ; b ) : b ∈ B } consistent with p ( x ). Definition 1.1.
Fix ℓ ∈ R . We say that a formula ϕ ( x ; y ) has VC-density ≤ ℓ with respect to p if there exists K ∈ R such that, for allfinite B ⊆ C y | S ϕ ( B ) ∩ [ p ] | ≤ K · | B | ℓ . If p ( x ) is the partial type x = x , we drop “with respect to p .”One interesting open question, originally posed in [1], is can we de-termine the VC-density of formulas in many variables knowing theVC-density of formulas in one variable? Open Question 1.2.
If there exists k < ω such that, for all ϕ ( x ; y ) with | x | = 1 , the VC-density of ϕ ( x ; y ) is ≤ k , then does there exists afunction f : ω → ω such that, for all ϕ ( x ; y ) , the VC-density of ϕ is ≤ f ( | x | ) ? N VC-DENSITY OVER INDISCERNIBLE SEQUENCES 3
Another question is can we relate dp-rank and VC-density in a nat-ural way?
Open Question 1.3.
Fix n < ω . Is it true that p ( x ) has dp-rank ≤ n if and only if, for all formulas ϕ ( x ; y ) , ϕ has VC-density ≤ n withrespect to p ? In particular, is it true that T is dp-minimal if and only if the VC-density of any formula ϕ ( x ; y ) is ≤ | x | ? One direction is clear, namely: Proposition 1.4.
Fix a partial type p ( x ) and n < ω . If, for all formu-las ϕ ( x ; y ) , ϕ has VC-density ≤ n with respect to p , then p has dp-rank ≤ n . This is a obvious generalization of the proof of Proposition 3.2 of[3]. These open questions are difficult to answer in this general setting.However, if we restrict ourselves to indiscernible sequences, we cananswer both of these questions.
Definition 1.5.
Fix ℓ ∈ R . We say that a formula ϕ ( x ; y ) has VC ind -density ≤ ℓ with respect to p if there exists K ∈ R such that, for allfinite indiscernible sequences b = h b i : i < N i , | S ϕ ( B ) ∩ [ p ] | ≤ K · N ℓ , where B = { b i : i < N } .In the remainder of this paper, we prove the following theorems. Theorem 1.6. VC ind -density is integer valued. This answers an open question posed by Aschenbrenner, Dolich,Haskell, MacPherson, and Starchenko in [1].
Theorem 1.7.
For any n < ω , a partial type p ( x ) has dp-rank ≤ n ifand only if all formulas ϕ ( x ; y ) have VC ind -density ≤ n with respect to p . This answers Open Question 1.3 for VC ind -density. As a corollary ofTheorem 1.7 and Theorem 4.8 of [8], we also answer Open Question1.2 for VC ind -density.
Corollary 1.8.
Fix k < ω and suppose that, for all ϕ ( x ; y ) with | x | =1 , the VC ind -density of ϕ ( x ; y ) is ≤ k . Then, for all ϕ ( x ; y ) , the VC ind -density of ϕ is ≤ k · | x | . VINCENT GUINGONA, CAMERON DONNAY HILL Local dp-rank
In order to prove Theorem 1.6 and Theorem 1.7, we introduce anotion of local dp-rank, which is just the obvious localization of dp-rank. This will turn out to be exactly equal to VC ind -density.In this section, we borrow notation from Chapters 3 and 4 of [18].Let ( I ; < ) be a linear order and let compl( I ) denote the completionof I . For any C ⊆ compl( I ), let ∼ C be the equivalence relation on I defined as follows. For i, j ∈ I , i ∼ C j if and only if, for all c ∈ C , wehave that c ≤ i ⇔ c ≤ j and i ≤ c ⇔ j ≤ c . Notice that, for any C ,this is a convex equivalence relation on ( I ; < ). Definition 2.1.
Fix a partial type p ( x ) and n < ω . We say that p has dp-rank ≤ n if, for all a (cid:15) p and all indiscernible sequences b = h b i : i ∈ Q i , there exists C ⊆ R with | C | ≤ n such that, for all i, j ∈ Q with i ∼ C j , tp( b i /a ) = tp( b j /a ).This definition is equivalent to the standard definition of dp-rankusing ICT-patterns by Proposition 4.20 of [18], which is a simple gen-eralization of of Lemma 1.4 of [17]. This is also equivalent to thedefinition of dp-rank using mutually indiscernible sequences, as in [8].We turn now to a localization of this definition.Let p ( x ) be any partial type and let ϕ ( x ; y ) be any formula. Definition 2.2.
For some n < ω , we say that the (local) dp-rank of ϕ with respect to p is ≤ n if, for all a (cid:15) p and all indiscernible sequences b = h b i : i ∈ Q i in C y , there exists C ⊆ R with | C | ≤ n such that, i, j ∈ Q with i ∼ C j , (cid:15) ϕ ( a ; b i ) ↔ ϕ ( a ; b j ).This local dp-rank is the correct localization of dp-rank in the fol-lowing sense. Proposition 2.3.
A partial type p ( x ) has dp-rank ≤ n if and only if,for all formulas ϕ ( x ; y ) , ϕ has local dp-rank ≤ n with respect to p .Proof. Suppose p has dp-rank ≤ n . Fix ϕ ( x ; y ), a (cid:15) p , and b = h b i : i ∈ Q i indiscernible. By definition, there exists C ⊆ R with | C | ≤ n so that, for all i, j ∈ Q with i ∼ C j , tp( b i /a ) = tp( b j /a ). In particular, (cid:15) ϕ ( a ; b i ) ↔ ϕ ( a ; b j ).Suppose that, for all formulas ϕ ( x ; y ), ϕ has local dp-rank ≤ n withrespect to p . Fix a (cid:15) p , and b = h b i : i ∈ Q i indiscernible. Then, foreach ϕ ( x ; y ), there exists a C ϕ ⊆ R with | C ϕ | ≤ n such that, i, j ∈ Q with i ∼ C ϕ j , (cid:15) ϕ ( a ; b i ) ↔ ϕ ( a ; b j ). Suppose that C ϕ is minimal such,and it is easy to see that such a set is unique. Let C = S ϕ ( x ; y ) C ϕ .Clearly, for all i, j ∈ Q with i ∼ C j , we have tp( b i /a ) = tp( b j /a ). N VC-DENSITY OVER INDISCERNIBLE SEQUENCES 5
Therefore, it suffices to show that | C | ≤ n . By means of contradiction,suppose | C | > n . In particular, there exists ϕ ( x ; y ) , ..., ϕ n ( x ; y ) so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ k ≤ n C ϕ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > n. Let ψ ( x ; y , y ) = W k ≤ n [ ϕ k ( x ; y ) ↔ ¬ ϕ k ( x ; y )]. There exists a subse-quence b ′ of pairs of b , indexed by ( n + 1) × Z ordered lexicographically,so that for k ≤ n and m ∈ Z , (cid:15) ψ ( a ; b ′ k,m ) if and only if m = 0. Usingcompactness, this shows that ψ has local dp-rank > n with respect to p , contrary to assumption. (cid:3) We say that a theory is strongly dependent if, for all partial types p ( x ), the dp-rank of p is finite (see [15] or Section 4.3 of [18]). Thereare theories that are dependent but not strongly dependent. However,a formula ϕ ( x ; y ) is dependent if and only if it has finite dp-rank withrespect to x = x . We use an example to illustrate this differencebetween local and global definitions. Example . Consider the language L = { E i : i < ω } with countablymany binary relations and let T be the L -theory where each E i isan equivalence relation and the intersection of any number of classesis infinite. Clearly this theory is dependent (in fact, stable), but it isnot strongly dependent. However, any formula has finite local dp-rank.The issue is that there are formulas with arbitrarily large local dp-rank;for example, consider ϕ n ( x ; y , y , ..., y n − ) = ^ i Before we show that local dp-rank and VC ind -density coincide, weintroduce another rank which is an indiscernible version of UDTFS-rank (see, for example, [4], [5], or [1]). Fix p ( x ) a partial type and ϕ ( x ; y ) a formula. For some B ⊆ C y , q ( x ) ∈ S ϕ ( B ) ∩ [ p ], and formula ψ ( y ), we say that ψ defines q ( x ) if, for all b ∈ B , (cid:15) ψ ( b ) if and only if ϕ ( x ; b ) ∈ q ( x ). Definition 3.1. For n < ω , we say that ϕ has UDTFS ind -rank ≤ n with respect to p if there exists a finite set of formulasΨ = { ψ r ( y ; z , , ..., z ,ℓ , ..., z n, , ..., z n,ℓ , w , ..., w k ) : r < R } such that, for all finite indiscernible sequences b = h b i : i < N i , thereexists j , ..., j k < N such that, for all q ( x ) ∈ S ϕ ( b ) ∩ [ p ], there exists VINCENT GUINGONA, CAMERON DONNAY HILL i , ..., i n < N and r < R so that ψ r ( y ; b i , ..., b i + ℓ , ..., b i n , ..., b i n + ℓ , b j , ..., b j k ) defines q ( x ) . Lemma 3.2. If ϕ has UDTFS ind -rank ≤ n with respect to p , then ithas VC ind -density ≤ n with respect to p .Proof. Under the hypothesis, a simple count reveals that | S ϕ ( B ) ∩ [ p ] | ≤ L · | B | n (where B = { b i : i < N } ). (cid:3) We now show that local dp-rank, VC ind -density, and UDTFS ind -rankcoincide. Theorem 3.3. For a formula ϕ ( x ; y ) , a partial type p ( x ) , and n < ω ,the following are equivalent: (1) ϕ has local dp-rank ≤ n with respect to p . (2) ϕ has UDTFS ind -rank ≤ n with respect to p . (3) ϕ has VC ind -density ≤ n with respect to p . (4) ϕ has VC ind -density ≤ ℓ with respect to p for some ℓ ∈ R with n ≤ ℓ < n + 1 .Proof. (1) ⇒ (2): Suppose ϕ has local dp-rank ≤ n with respect to p . Claim. There exists ℓ < ω such that, for any finite indiscerniblesequence b = h b i : i < N i and all a (cid:15) p , there exists i < ... < i n < N such that, for all k ≤ n and all j, j ′ with i k + ℓ < j < j ′ < i k +1 , (cid:15) ϕ ( a ; b j ) ↔ ϕ ( a ; b j ′ ) (for k = 0 , n , let i + ℓ = − i n = N ). Proof of Claim. Suppose not. Then, for each ℓ < ω , we can find some b = h b i : i < N i and a (cid:15) p for which no such i < ... < i n < N exist. A simple compactness argument then shows that we can buildan indiscernible sequence b = h b i : i ∈ Q i and a (cid:15) p which witnessesthe fact that ϕ has dp-rank > n with respect to p . (cid:3) From the claim it is easy to explicitly show ϕ has UDTFS ind -rank ≤ n with respect to p . As in the proof of Theorem 1.2 (ii) of [4], thereexists finitely many formulas θ s ( y , y ; w , ..., w k ) for s < S such that,for any finite indiscernible sequence b , either b is a indiscernible set orthere exists j , ..., j k and s < S so that θ s ( y , y ; b j , ..., b j k ) defines theindiscernible ordering of b (i.e., h b i , b i ′ i holds of this formula if and onlyif i < i ′ ).We need now a formula for each possible configuration of truth valuesbetween a finite indiscernible sequence b and a (cid:15) p . Consider the set X = ( { } × { , ..., n } ) ∪ ( { , ..., n } × { , ..., ℓ } )and, for each f : X → { , } and s < S , define the formula ψ f,s ( y ; z , , ..., z ,ℓ , ..., z n, , ..., z n,ℓ , w , ..., w k ) N VC-DENSITY OVER INDISCERNIBLE SEQUENCES 7 which satisfies:(i) If y = z i,j , then ψ f,s holds iff. f ( i, j ) = 1,(ii) If y falls in between z i,ℓ and z i +1 , via the ordering given by θ s ( y , y ; w , ..., w k ), then ψ f,s holds iff. f (0 , i ) = 1.These formulas take care of the case where we are dealing with anindiscernible sequence that is not an indiscernible set. For the othercase, we define formulas ψ ∗ f ( y ; z , ..., z nℓ ) for each f : { , , ..., nℓ } →{ , } so that(i) For i > 0, if y = z i , ψ ∗ f holds iff. f ( i ) = 1,(ii) For y = z i for all i > ψ ∗ f holds iff. f (0) = 1.Now, for any finite indiscernible sequence b and a (cid:15) p , let i < ... n , witnessed by a and h b i : i ∈ Q i . Choose C ⊆ R minimal so that, for all i, j ∈ Q with i ∼ C j , (cid:15) ϕ ( a ; b i ) ↔ ϕ ( a ; b j ), so | C | > n . If C is infinite, ϕ ( x ; y ) has IPwith respect to p , hence ϕ ( x ; y ) has infinite VC ind -density, so we mayassume C is finite. Therefore, there exists at least n + 2 infinite ∼ C classes, call them D < D < ... < D n +1 . For each consecutive pair D ℓ and D ℓ +1 , either the truth value of ϕ ( a ; b i ) differs or there exists some j ℓ ∈ Q with D ℓ < j ℓ < D ℓ +1 on which the truth value of ϕ ( a ; b j ℓ ) differsfrom both. If the truth values on D ℓ and D ℓ +1 differ, choose j ℓ ∈ D ℓ +1 arbitrarily. Now fix N < ω and i = h i , ..., i n i an ( n + 1)-tuple from N with i < ... < i n and let B N = { b i : i = 0 , , ..., N − } . Choose anorder-preserving map g i : N → Q so that(i) g i ( i ℓ ) = j ℓ , and(ii) If i ℓ − < i < i ℓ , then g i ( i ) ∈ D ℓ .Then consider the indiscernible subsequence h b g ( i ) : i < N i (for g = g i ).By indiscernibility, there exists a i (cid:15) p so that, for all i = 0 , ..., N − (cid:15) ϕ ( a i ; b i ) ↔ ϕ ( a ; b g ( i ) ) . VINCENT GUINGONA, CAMERON DONNAY HILL However, by the choice of D ℓ and j ℓ , each type tp ϕ ( a i /B N ) is distinctfor different choices of i . Therefore, | S ϕ ( B N ) ∩ [ p ] | ≥ (cid:18) Nn + 1 (cid:19) , which is on the order of N n +1 . Hence, the VC ind -density of ϕ is ≥ n +1.Therefore, for each ℓ < n + 1, VC ind -density of ϕ is > ℓ . This is whatwe aimed to prove. (cid:3) Notice that Theorem 3.3 (3) ⇔ (4) provides a proof of Theorem 1.6.Moreover, Theorem 3.3 (1) ⇔ (3) and Proposition 2.3 immediately givea proof of Theorem 1.7. Consider the following theorem. Theorem 3.4 (Theorem 4.8 of [8]) . Suppose A is a set, a , a aretuples, and k , k < ω . If tp( a i /A ) has dp-rank ≤ k i for i = 1 , , then tp( a , a /A ) has dp-rank ≤ k + k . This, coupled with Theorem 1.7 gives a proof of Corollary 1.8. As aresult of this corollary, we get the following. Corollary 3.5. A theory T is dp-minimal if and only if all formulas ϕ ( x ; y ) have VC ind -density ≤ | x | . Future Directions It is our hope that techniques developed in this paper for VC ind -density can me modified to answer the open questions about VC-density.The definition of UDTFS ind -rank above differs drastically from theoriginal definition of UDTFS-rank given in [5] (related to the VC d property in [1]). Fix a partial type p ( x ) and a formula ϕ ( x ; y ). Wesay that ϕ has UDTFS-rank ≤ n with respect to p ( x ) if there existsfinitely many formulas ψ r ( y ; z , ..., z n ) for r < R such that, for everyfinite B ⊆ C y and q ( x ) ∈ S ϕ ( B ) ∩ [ p ], there exists b , ..., b n ∈ B and r < R such that ψ r ( y ; b , ..., b n ) defines q ( x ) . It is easy to show that the VC-density of ϕ with respect to p is boundedby the UDTFS-rank of ϕ with respect to p (by a simple counting ar-gument). Moreover, the following holds. Proposition 4.1 (Theorem 3.13 of [5], Theorem 5.7 of [1]) . If there ex-ists k < ω such that, for all formulas ϕ ( x ; y ) with | x | = 1 , the UDTFS-rank of ϕ is ≤ k , then, for all formulas ϕ ( x ; y ) , the UDTFS-rank of ϕ is ≤ k · | x | . N VC-DENSITY OVER INDISCERNIBLE SEQUENCES 9 However, it is easy to show that VC-density and UDTFS-rank donot coincide, so Proposition 4.1 cannot be directly applied to answerOpen Question 1.2. Can we find a new rank by modifying UDTFS-rank to make it look similar to UDTFS ind -rank such that the followingtwo conditions hold:(i) this rank is subadditive as in Proposition 4.1, and(ii) this rank bounds VC-density and vice versa?If such a rank exists, this would settle Open Question 1.2. We cannothope for a (integer valued) rank that coincides exactly with VC-density,since VC-densiy can be non-integer valued. For example, see Proposi-tion 4.6 of [1].Another interesting question arising from our investigation is thatof when VC-density and VC ind -density coincide. It is very easy to seethat VC ind -density is bounded by VC-density, but it is not at all clearhow one might use VC ind -density to estimate or bound the generalVC-density. These two values certainly do not coincide in general, forVC ind -density is always integer-valued, while VC-density can take non-integer values even in very simple scenarios. Quite recently, however,Johnson (see Corollary 3.5 of [7]) has demonstrated a necessary andsufficient condition for the VC-density and VC ind -density of a givenformula to be equal. References [1] M. Aschenbrenner, A. Dolich, D. Haskell, H.D. MacPherson, and S.Starchenko, Vapnik-Chervonenkis density in some theories without the inde-pendence property, I . preprint.[2] M. Bodirsky, New Ramsey classes from old (2012). preprint.[3] A. Dolich, J. Goodrick, and D. Lippel, dp-Minimality: Basic facts and exam-ples , Notre Dame J. Form. Log. (2011), no. 3, 267–288.[4] V. Guingona, On uniform definability of types over finite sets , J. SymbolicLogic (2012), no. 2, 499–514.[5] V. Guingona and M.C. Laskowski, On VC-minimal theories and variants ,Archive for Mathematical Logic. to appear.[6] C.D. Hill, Generalized indiscernibles as model-complete theories (2012).preprint.[7] H. Johnson, Vapnik-Chervonenkis density on indiscernible sequences, stability,and the maximum property (2013). preprint.[8] I. Kaplan, A. Onshuus, and A. Usvyatsov, Additivity of the dp-rank . preprint.[9] A. S. Kechris, V. G. Pestov, and S. Todorcevic, Fraisse limits, Ramsey the-ory, and topological dynamics of automorphism groups , Geom. Funct. Anal. (2005), 106-189.[10] M.C. Laskowski, Vapnik-Chervonenkis Classes of Definable Sets , J. LondonMath. Soc. (1992), no. 2, 377–384. [11] M. Malliaris and S. Shelah, Regularity lemmas for stable graphs , Trans. Amer.Math. Soc.[12] S. Shelah, Classification theory and the number of non-isomorphic models ,North-Holland Publishing Company, 1978.[13] , Classification theory for elementary classes with the dependence prop-erty - a modest beginning , Scientiae Math Japonicae (2004), no. 2, 265–316.[14] , Dependent first order theories, continued , Israel J Math (2009),1–60.[15] , Strongly dependent theories (2009). preprint.[16] , Dependent theories and the generic pair conjecture (2012). preprint.[17] P. Simon, On dp-minimal ordered structures , J. Symbolic Logic (2011),no. 2, 448–460.[18] , Lecture notes on NIP theories , 2012. preprint.(Guingona) University of Notre Dame, Department of Mathematics,255 Hurley, Notre Dame, IN 46556 E-mail address : [email protected] URL : (Hill) Wesleyan University, Department of Mathematics and Com-puter Science, 45 Wyllys Avenue, Middletown, CT 06459 E-mail address ::