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Proof. This follows easily from Lemma 5.1.17 of Buechler [7], which states that the indiscernible sequence I is a Morley sequence over C ∪J for any infinite J ⊂ I. To apply this, let i 1 , . . . , i n ∈ I be distinct, and let J 1 , J 2 be infinite disjoint subsets of Stably embedded sets.A C-definable set D in U is stably embedded if, for any definable set E and r > 0, E ∩ D r is definable over C ∪ D. If instead we worked in a small model M , and C, D were from M eq , we would say that D is stably embedded if for Proof. (i) We may suppose that A st | ⌣C M st . Then, by the hypothesis, tp(A st A/M ) is the unique extension of tp(A st A/C) ∪ tp(A st /M ) to M . Since tp(A st /M st ) is definable over acl(C) in the stable structure St C , it follows from Lemma 3.12 that tp(A/M ) is definable over acl(C). In particular, tp(A/M ) is Aut(M/acl(C))-invariant.(ii) Now suppose A ′′ ≡ acl(C) A ′ and tp(A ′ /M ), tp(A ′′ /M ) are both Aut(M/acl(C))invariant extensions of tp(A/C). Then tp((A ′ ) st /M st ) and tp((A ′′ ) st /M st ) are both invariant extensions of tp(A st /acl(C)) in St C : indeed, any automorphism in Aut(St C /acl(C)) extends to an automorphism of M over acl(C) (saturation of M and stable embeddedness), so fixes tp(A ′ /M ) and tp(A ′′ /M ), and hence fixes tp((A ′ ) st /M st ) and tp((A ′′ ) st /M st ). Hence tp((A ′ ) st /M st ) and tp((A ′′ ) st /M st ) are equal, as invariant extensions of a type over an algebraically closed base are

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and P -minimal theories.Our approach to Theorem 1.1, via definable types, was partly inspired by the use of Puiseux series in [11,81].Let ACVF denote the theory of (non-trivially) valued algebraically closed fields, in the ring language expanded by a predicate for the valuation divisibility. This has completions ACVF (0,0) (for residue characteristic 0), ACVF (0,p) (field characteristic 0, residue characteristic p), and ACVF (p,p) (field characteristic p). Because ACVF (0,0) is interpretable in RCVF, our methods give (non-optimal) density bounds for ACVF (0,0) (Corollary 6.3). However, they give no information on density in the theories ACVF (0,p) and ACVF (p,p) . The problems arise essentially because a definable set in 1-space in ACVF is a finite union of 'Swiss cheeses' but we have no way of choosing a particular Swiss cheese. This means that the definable types technique in our main tool (Theorem 5.7) breaks down. On the other hand, our methods do yield:Theorem 1.2. Suppose M = Q p is the field of p-adic numbers, construed as a firstorder structure in Macintyre's language L p . Then the VC density of every L p -formula ϕ(x; y) is at most 2|y| − 1.

In this paper we formulate a notion similar to o-minimality but appropriate for the p-adics. The paper is in a sense a sequel to [11] and [5]. In [11] a notion of minimality was formulated, as follows. Suppose that L, L+ are first-order languages and + is an L+-structure whose reduct to L is . Then + is said to be -minimal if, for every N+ elementarily equivalent to +, every parameterdefinable subset of its domain N+ is definable with parameters by a quantifier-free L-formula. Observe that if L has a single binary relation which in is interpreted by a total order on M, then we have just the notion of strong o-minimality, from [13]; and by a theorem from [6], strong o-minimality is equivalent to o-minimality. If L has no relations, functions, or constants (other than equality) then the notion is just strong minimality.In [11], -minimality is investigated for a number of structures . In particular, the C-relation of [1] was considered, in place of the total order in the definition of strong o-minimality. The C-relation is essentially the ternary relation which naturally holds on the maximal chains of a sufficiently nice tree; see [1], [11] or [5] for more detail, and for axioms. Much of the motivation came from the observation that a C-relation on a field F which is preserved by the affine group AGL(1,F) (consisting of permutations (a,b) : x ↦ ax + b, where a ∈ F \ {0} and b ∈ F) is the same as a non-trivial valuation: to get a C-relation from a valuation ν, put C(x;y,z) if and only if ν(y − x) < ν(y − z).

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