I-Indexed Indiscernible Sets and Trees

Transcription

1 1 / 20 I-Indexed Indiscernible Sets and Trees Lynn Scow Vassar College Harvard/MIT Logic Seminar December 3, 2013

2 2 / 20 Outline 1 background 2 the modeling property 3 a dictionary theorem

3 3 / 20 order indiscernible sets Fix a linear order O and an L-structure M (we assume M is sufficiently saturated.) Let b i be same-length finite tuples from M: Definition B = {b i i O} is an order-indiscernible set if for all n 1, for all i 1,..., i n, j 1,..., j n from O, (i 1,..., i n ) (j 1,..., j n ) is an order-isomorphism tp L (b i1,..., b in ; M) = tp L (b j1,..., b jn ; M)

4 4 / 20 typical application Suppose we have parameters A = {a i i < ω} and i < j ϕ(a i, a j ) (let s assume ϕ(x, x) is unsatisfiable) In a typical application, we use Ramsey s theorem to find an order-indiscernible set B = {b i i < ω} such that i < j ϕ(b i, b j ) Because B is indiscernible, for some t {0, 1} (ϕ 0 = ϕ, ϕ 1 = ϕ) i > j ϕ(b i, b j ) t In a well-known characterization: Th(M) is stable t = 0 for all such B

5 5 / 20 generalizing order-indiscernible sets Consider O as a structure in its own right, O = (O, <) in the language L = {<}, and re-write the definition: Definition B = {b i : i O} is an order-indiscernible set if for all n 1, for all i 1,..., i n, j 1,..., j n from O, (i 1,..., i n ) (j 1,..., j n ) tp L (b i1,..., b in ; M) = tp L (b j1,..., b jn ; M) Here (i 1,..., i n ) (j 1,..., j n ) means qftp L (i 1,..., i n ; O) = qftp L (j 1,..., j n ; O)

6 6 / 20 I-indexed indiscernible sets Now we fix an arbitrary language L, and an L -structure I in the place of O. Definition ([She90]) B = {b i : i I} is an I-indexed indiscernible set if for all n 1, for all i 1,..., i n, j 1,..., j n from I, (i 1,..., i n ) (j 1,..., j n ) tp L (b i1,..., b in ; M) = tp L (b j1,..., b jn ; M) Here (i 1,..., i n ) (j 1,..., j n ) means qftp L (i 1,..., i n ; I) = qftp L (j 1,..., j n ; I) Say that B is -I-indexed indiscernible for L if we replace L above by.

7 7 / 20 overview Suppose ϕ(x, x) is unsatisfiable. Then the type of a {ϕ}-i-indexed indiscernible set B is determined entirely by the data t = (t 0, t 1,...) If B is an order-indiscernible set: i < j ϕ(b i, b j ) t0 i > j ϕ(b i, b j ) t1 If B is an ordered-graph indexed indiscernible set i < j irj ϕ(b i, b j ) t0 i > j irj ϕ(b i, b j ) t1 i < j irj ϕ(b i, b j ) t2 i > j irj ϕ(b i, b j ) t3

8 8 / 20 ordered graphs Consider the example I = (I, <, E) for an order relation < and an edge relation E. Suppose we only consider I that are weakly saturated, i.e., that embed all possible ordered graphs. The above kind of I-indexed indiscernible can be applied to characterize NIP theories. We call it an ordered graph-indiscernible set. Suppose we have an ordered graph-indexed set B such that T is NIP t = 0 for all such B i < j irj ϕ(b i, b j ) i < j irj ϕ(b i, b j ) t In a characterization from [Sco12]: T is NIP iff any ordered graph indiscernible set in a model of T is an order-indiscernible set.

9 9 / 20 different partition properties Fix a coloring on n-tuples from I, where coloring is uniform on pairs: large homogeneous B I s.t. (i, j) from B: irj irj i < j red blue i > j green purple irj irj i < j r (b) r (b) i > j p (g) p (g) (Ramsey s theorem) irj irj i < j red blue i > j green purple (Nešetřil-Rödl theorem)

10 10 / 20 trees I s = (ω <ω,,, < lex, (P n ) n<ω ) where is the partial tree-order, is the meet function in this order, < lex is the lexicographical order, and the P n are predicates picking out the n-th level of the tree I 1 = (ω <ω,,, < lex, < lev ) where η < lev ν l(η) < l(ν) I 0 = (ω <ω,,, < lex ) I t = (ω <ω,, < lex )

11 11 / 20 a typical dichotomy result The structure I s is ideal to study TP Definition A theory T has the (2-)tree property (TP) if there is a model M T, a formula ϕ(x; y) and parameters a η from M with l(a η ) = l(y) such that: 1 {ϕ(x; a σ n ) : σ ω ω } is consistent (nodes on a path are consistent ), and 2 for all η ω <ω, pairs from {ϕ(x; a η i ) : i < ω} are inconsistent (siblings are inconsistent ) By a well-known result, if a theory has TP, then it has TP as witnessed by B = {b η η ω> ω} where B is I s -indexed indiscernible. By a series of reductions, one proves the well-known theorem that TP comes in one of two extremal versions...tp1 and TP2.

12 12 / 20 ramsey classes: I Fix a class K of finite L -structures. Definition For A, B K, a copy of A in B is an embedding f : A B modulo the equivalence relation of being the same embedding up to an automorphism of A From now on, assume L contains a relation < linearly ordering all members of K. Then we may think of a copy of A in B as being the range of an embedding from A into B. We denote the copies of A in B as ( B A).

13 13 / 20 ramsey classes: II Given a finite set X of cardinality k, We refer to a map c : ( C A) X as a k-coloring of the copies of A in C. We say that B C is homogeneous for c if there is an element x 0 X such that for all A ( B A ), c(a ) = x 0. Definition A class K of finite L -structures is a Ramsey class if for all A, B K there is a C K such that for any 2-coloring of ( C A), there is a B C, isomorphic to B that is homogeneous for this coloring.

14 14 / 20 EM-types For A = {a i i I} we can formally define a type in variables {x i i I} called the Ehrenfeucht-Mostowski type of A, EM(A) If ϕ(a i1,..., a in ) for all (i 1,..., i n ) (j 1,..., j n ), then ϕ(x j1,..., x jn ) EM(A) If B EM(A), and q is a complete quantifier free type in the language of I, then if then ı (q(ı) ϕ(a i )) ı ( q(ı) ϕ(b i ) ) In fact B will have a rule such as the above for all quantifier-free types q ; whereas A could have rules for none.

15 15 / 20 the modeling property Definition I-indexed indiscernibles have the modeling property if for all I-indexed parameters A = (a i : i I) in any structure M, there exists I-indexed indiscernible parameters B EM(A) For which I do I-indexed indiscernibles have the modeling property?

16 16 / 20 translation Theorem (dictionary theorem) Suppose that I is a qfi, locally finite structure in a language L with a relation < linearly ordering I. Then I-indexed indiscernible sets have the modeling property just in case age(i) is a Ramsey class. Recall I 0 = (ω <ω,,, < lex ) Theorem (Takeuchi-Tsuboi) I 0 -indexed indiscernibles have the modeling property. Corollary age(i 0 ) is a Ramsey class. Removing destroys the Ramsey property.

17 17 / 20 K = age(i t ) not a Ramsey class Proof. By [Neš05], if K is a Ramsey class, then K has the amalgamation property. However, an example analyzed in Takeuchi-Tsuboi provides a counterexample to amalgamation. A L t -embeds into B 1, B 2 by a i b i, c i. a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 A: B 1 : b 4 B 2 : c 4 a 0 b 0 c 0 Suppose there exists some amalgam C for (A, B 1, B 2 ). Observe that b 4, c 4 in C must be -comparable in C, as both points are -predecessors of the same point, b 2 (= c 2 ). If b 4 c 4, then b 4 c 4 c 3 = b 3, contradicting the data in B 1. If c 4 b 4, then c 4 b 4 b 1 = c 1, contradicting the data in B 2.

18 18 / 20 finitary infinitary Theorem ([She90]) For every n, m < ω there is some k = k(n, m) < ω such that for any infinite cardinal χ, the following is true of λ := ℶ k (χ) + : for every f : ( n λ) m χ there is a level-preserving, orientation-preserving subtree I n λ such that (i) I and whenever η I n> λ, {α < λ : η α I} χ +. (ii) f If η, ν I are such that η Is ν then f(η 0,..., η m 1 ) = f(ν 0,..., ν m 1 ). Theorem ([Fou99]) age(i s ) is a Ramsey class Both yield that I s -indexed indiscernibles have the modeling property, the second by way of the dictionary theorem. The first result yields a height-n indiscernible subtree with m-types from the original tree.

19 19 / 20 Thanks Thanks for your attention!

20 20 / 20 W. L. Fouché. Symmetries and Ramsey properties of trees. Discrete Mathematics, 197/198: , th British Combinatorial Conference (London, 1997). J. Nešetřil. Homogeneous structures and Ramsey classes. Combinatorics, Probability and Computing, 14: , L. Scow. Characterization of NIP theories by ordered graph-indiscernibles. Annals of Pure and Applied Logic, 163: , S. Shelah. Classification Theory and the number of non-isomorphic models (revised edition). North-Holland, Amsterdam-New York, 1990.