AN ABSTRACT APPROACH IN CANONIZING TOPOLOGICAL RAMSEY SPACES 7 1 DIMITRISVLITAS 0 2 n Abstract. In[To]S.TodorcevicintroducedthenotionofatopologicalRam- a sey space and a list of axioms required to be satisfied by any such a space. J HereweshowthatanytopologicalRamseyspacethatsatisfiesastrengthened 6 version of one of the requiredaxioms and a very natural assumption, admits 2 canonization theorem. ] O L . h 1. Introduction t a In[To]S.TodorcevicintroducedthenotionofatopologicalRamseyspace. Topo- m logical Ramsey spaces are structures of the form hR,≤,r,i, satisfying certain con- [ ditions (see in the next section). Someone should think of R as family of infinite 1 sequence of objects and the function r as finite approximations of these infinite v 6 sequences. Then ≤ is a quasi-ordering on R. These spaces admit a natural topol- 6 ogy and they are required to satisfy four axioms, A.1−A.4. As a consequence of 5 these axioms one gets that Ramsey subsets of R are exactly those with the Baire 7 0 property and meager sets are Ramsey null. Many of the well known spaces can be 1. seen as instances of topological Ramsey spaces. The most well known example is 0 the Ellentuck space hN[∞],⊆,ri. For A ∈ N[∞], r (A) is the initial segment of A n 7 formed by taking the first n-elements of A. 1 : Canonical results in Ramsey theory try to describe equivalence relations in a v given Ramsey structure, based on the underlying pigeonhole principles. The first i X example of them is the classical Canonization Theorem by P. Erdo˝s and R. Rado r a [Er-Ra] which can be presented as follows: Given α≤β ≤ω let β :={f(α) : f :α→β is strictly increasing}. α (cid:18) (cid:19) The previous is commonly denoted by [β]α. Then for any n < ω and any finite coloring of ω there is an isomorphic copy M of ω (i.e. the image of a strictly n increasing f : ω → ω) and some I ⊆ n(:= {0,1,...,n−1}) such that any two n- (cid:0) (cid:1) element subsets have the same color if and only if they agree on the corresponding relative positions given by I. This was extended by P. Pudla´k and V. Ro¨dl in [Pu-Ro] for colorings of a given uniform family G of finite subsets of ω by showing that given any coloring of G, thereexistsAaninfinite subsetofω,auniformfamilyT andamappingf :G →T 1 2 DIMITRISVLITAS such that f(X) ⊆ X for all X ∈ G and such that any two X,Y ∈ G ↾ A have the same color if and only if f(X)=f(Y). The P. Erdo˝s-Rado result deals with equivalence relations on the family of k approximations of elements of members of N[∞]. The Pudla´k-Rödl result deals with equivalence relations on uniform families of finite approximations of elements of members of N[∞]. In this paper we are going to generalize the above results to anytopologicalRamseyspace. Namelythatgivenafamilyoffiniteapproximations F ofR,(seeDefinition1)andanequivalencerelationf :F →ω onit,thereexists an X ∈ R and a map φ, ( see Definition 2 ) so that for any s,t ∈ AX, s,t ∈ F it holds that f(s)=f(t) if and only if φ(s)=φ(t). 2. Background material Topological Ramsey spaces are spaces of the form hR,≤,ri, where r :R×ω → ARis a mapthatgivesus the sequencer(·,n)=r (·) ofapproximationmappings. n The basic open sets are of the form: [s,X]={Y ≤X :(∃n)r (Y)=s}, n for s ∈ AR and X ∈ R. If r (X) = s we write s ⊑ X. The axioms required to n be satisfied by any such a space in order to be topological Ramsey space are the following. A.1. Let X,Y ∈R. (1) r (X)=∅ for all X ∈R. 0 (2) X 6=Y implies r (X)6=r (Y) for some n∈ω. n n (3) r (X)=r (Y) implies n=m and r (X)=r (Y) for all k k. As a consequence to get X(n) we have used the levels U(k),...,U(l−1) of U. For s∈AU, with depth (s)=k, we define U X[s]={t∈AU :s⊑t and t≤ r (Y),for some Y ≤X,j <ω}. fin j Observe that X[s] 6= ∅ if and only if [s,X] 6= ∅. In this case we say that X is compatible with s. Notice also that for every t∈U[s] there exists a set X ⊆[k,l), where depth (t)=l, so that t\s is made out of ∪ U(n). U n∈X Next given s and X, so that [s,X]6=∅, by X/s we denote X\s. 3. Main theorem We introduce the notion of a Front. Definition 1. A family F of finite approximations of reducts of U is called a front, if for every X ≤U, there exists s∈F so that s⊑X and for any two distinct s,t∈F, is not the case that s⊑t. We distinguish a specific case of fonts the families of finite approximations of length n. Namely AU ={s:s=r (X),X ≤U}. n n Given a front F on [∅,X], X ≤U we introduce Fˆ defined as follows: Fˆ ={t∈AU :∃s∈F,t⊑s} observe that ∅∈Fˆ. For t∈Fˆ\F F ={s∈F :t⊑s}. t Notice that F is a front on U/t. For Y ≤X t F ↾Y ={t∈F :t∈AY′,Y′ ≤Y}, Fˆ ↾Y ={t∈Fˆ :t∈AY′,Y′ ≤Y}. Next we introduce a stronger version of A.4 as follows: A.4⋆ Let s∈AU , depth (s)=k, X ≤U with [s,X]6=∅ and a coloring n U c:[s,X] →ω, n+1 where [s,X] ={t∈AX :s⊑t}=r [s,X] be given. There exists a map n+1 n+1 n+1 φ :[s,U] →P((∪[s,U] ) [k,∞)) s n+1 n+1 so that φ (p) ⊆ p(n)∪[k,l), where l = depth (p), [and Y ∈ [s,X] so that for all s U p,q ∈ [s,Y] it holds that c(p) = c(q) if and only if φ (p) = φ (q). We will call n+1 s s such a mapping φ inner for s. s AN ABSTRACT APPROACH IN CANONIZING TOPOLOGICAL RAMSEY SPACES 5 Inotherwords,thereexistsareductY,wherethe coloringcdependents onlyon a subset of p(n), for any p∈[s,Y] , andthe levels U[k,l) needed to get p(n). In n+1 some sense φ gives us a subset of the information coded by p(n). s A.4⋆ essentially deals with the length one extensions of initial segments. This necessitatestointroduce the setofalllengthoneextensionsofallmembersofAX. Let LX ={w:∃s∈AX,s∪w ∈[s,X] }. 1 |s|+1 Observe that AX ⊆LX , cause r (X)=∅. Let 1 1 0 LX ={w:∃t∈AX,t∪w ∈[t,X] } n |t|+n and LX =∪ LX . n∈ω n Next we introduce a partial ordering on LX as follows. Given w,v ∈LX we write w ≤ v if there exists s ∈ AX so that s∪w∪v ∈ AX. Let w ,...,w elements of 0 n LX so that either w ≤···≤ w and ∃t ∈AX so that t∪w ∪···∪w ∈AX or 0 n 0 n w ∈X(n), for all i≤n, and ∃t∈AX such that t∪(w ,...,w )∈AX . Then i 0 n |t|+1 hw ,...w i ∈LX isdefinedtobethesetofallendextensionofs,s≤ t,made 0 n s 1 fin outof w ,...,w . When we say that the end extensionin made out of w ,...,w , 0 n 0 n we mean all w ,i ≤ n, are needed, not a proper subset is sufficient. Notice that if i s∪w∈[s,X] , then hwi =w in this case we say that hi acts trivially on w. |s|+1 s s To state the main theorem of this paper we need the following definition. Definition 2. Let F be a front on [∅,U] and let φ be a function on F. We call φ is Inner if for every t=t(i) ∈F we have that i∈n φ(t)=(φ (w ),...,φ (w )). s0 0 sm m Where φ ,...,φ are inner maps, for s = r (t),...,s = r (t) respec- s0 sm 0 h0 m hm tively, where h < ··· < h < n. Also w ∈ ht(i ),...,t(i )i ,...,w ∈ 0 m 0 0 l t(h0) m ht(j ),...,t(j )i , {i ,...,i ,...,j ,...,j ,h ,...,h } ⊆ n, and every t , 0 m t(hm) 0 l0 0 ld 0 m i i

An abstract approach in canonizing topological Ramsey spaces

Similar An abstract approach in canonizing topological Ramsey spaces