Universal graphs and functions on ω1

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Abstract

It is shown to be consistent with various values of b, d and 21 that there is a universal graph on ω1. Moreover, it is also shown that it is consistent that there is a universal graph on ω1 — in other words, a universal symmetric function from ω12 to 2 — but no such function from ω12 to ω. The method used relies on iterating well known reals, such as Miller and Laver reals, and alternating this with the PID forcing which adds no new reals. The last sections examine the question of set valued universal functions.

Introduction

It is well known that countably saturated models are universal for models of cardinality 1. However, in the case of graphs, the existence of a saturated model is equivalent to 20=1. These two observations immediately raise the question of whether it is possible to have a universal graph of cardinality 1 in the absence of the Continuum Hypothesis. This provided the motivation for the articles [1], [2] and [3] which solved not only this question, but also others about the universality of different structures.

While the results to be presented here have their roots in this work, they are also motivated by considerations that are not model theoretic. A function U:X2X is said to be (Sierpiński) universal if for any G:X2X there is e:XX such that G(x,y)=U(e(x),e(y)) for all x and y in X. The function e will be called an embedding of G into U. An early reference to this notion can be found in Problem 132 of the Scottish Book [4] in which Sierpiński asked if there is a Borel function which is universal in this sense, when X is the real line. He had already shown in [5] that there is a Borel universal function assuming the Continuum Hypothesis. This notion of universal function is also studied in Rado [6]. More recently, this notion and various generalizations of it were studied in [7], in which a restricted form of the following definition appears as Definition 7.5.

Definition 1.1

A function U:κ2λ is weakly universal if for every f:κ2λ there exist one-to-one functions h:κκ and k:λλ such that k(f(α,β))=U(h(α),h(β)) for all α and β in κ. The pair (h,k) will be called a weak embedding.

Definition 7.4 of [7] defines a function U:κ2κ to be model theoretically universal if it is weakly universal, as in Definition 1.1, but with h=k. Note that U is Sierpiński universal if it is weakly universal with k being the identity. Remark 7.7 of [7] claims that all three notions — Sierpiński universal, weakly universal, and model theoretically universal — are equivalent for maps into 2.

The following is Theorem 5.9 of [7] showing that there is no difference between asking about the existence of universal graphs — in other words, symmetric, irreflexive functions from ω12 to 2 — and non-symmetric functions from ω12 to 2.

Theorem 1.2

For any infinite cardinal κ the following are equivalent:

  • 1.

    For each nN there is a universal function from κ2 to n.

  • 2.

    For some nN with n2 there is a universal function from κ2 to n.

  • 3.

    There is a symmetric, irreflexive function from κ2 to 2 universal for all symmetric, irreflexive functions from κ2 to 2

  • 4.

    There is a universal graph on κ.

Given Theorem 1.2 it is reasonable to focus attention only on symmetric functions from ω12 to ω and this will be done from now on. This justifies the following:

Convention: Any function with domain ω12 will be assumed to be symmetric. In other words, f:ω12Y should be read as f:[ω1]2Y.

Moreover, given that there are only two possible values for k when λ=2, the validity of Remark 7.7 of [7] should now be clear and, of course, assuming the Continuum Hypothesis there is a Sierpiński universal function from R2 to R, and hence there is also a weakly universal function from ω12 to 2. Problem 7.8 of [7] asks if the existence of one type of universal function implies the existence of the others in general. To provide a negative answer to the question, it is therefore necessary to consider models where the Continuum Hypothesis fails. This provides an other path to the questions considered in [1], [2] and [3].

When κ=ω1 and λ=2 it was shown in [1] and [2] that it is consistent with the failure of the Continuum Hypothesis that there is a universal function U:ω122 which satisfies all three universality properties. The methods used in the generalization of this result by Mekler to other theories in [3] provide examples of Sierpinski universal functions U:ω12λ for λ equal to 2, ω or ω1. In the models of [1] and [2] the cardinal invariants b and d have the following values respectively: 1 and 2. One might, therefore ask, whether these values are needed for the existence of universal functions from U:ω12λ with the failure of the Continuum Hypothesis. This question becomes even more interesting in light of the positive results to be presented in Lemma 6.1 and Lemma 7.8 which rely on small values of b and d respectively. It will be shown in Corollary 3.10 that it is consistent with the existence of universal functions from U:ω122 that b=d=2 and in Corollary 4.19 that it is consistent with the existence universal functions from U:ω122 that b=d=1.

However, the main goal of this paper will be to answer Problem 5.10 of [7]. This is done in Corollary 6.2 which establishes that there is a Sierpiński universal function from ω12 to 2 but no such function from ω12 to ω. It is worth recalling that Theorem 5.9 of [7] asserts that if 2n<ω then there is a Sierpiński universal function from ω12 to 2 if and only if there is a Sierpiński universal function from ω12 to n. A solution to Problem 7.8 of [7] will also be presented under the assumption that d is small. This and related questions are discussed in §7. That section also contains some arguments extending the methods of [1] to some weak versions of embedding. Finally, if one is only interested in obtaining a model of set theory in which the Continuum Hypothesis fails, yet there is a universal graph of cardinality 1, then §2 presents an easier argument than the original of [1]. On the other hand, §9 presents a reformulation and simplification of arguments from [1].

The graphs that will be shown to be universal in the arguments to follow will all come from some initial model of set theory in which the Continuum Hypothesis holds. For §2, §3 and §4 the following will be relevant.

Definition 1.3

Given any function G:ω12ω and ηω1 define Gη:ηω by Gη(ζ)=G(ζ,η) and then define Sη(G)={Gμη}μη. A function G:ω12ω such that Sη(G) is everywhere non-meagre for each ηω will be called category saturated.

Definition 1.4

Let ν be an atomic probability measure on ω and let νη be the Fubini product of this measure on ωη for any ηω1. A function G:ω12ω such that Sη(G) has outer measure 1 for each ηω will be called ν-saturated. The notion defined here will not be needed in this full generality, but is does find an application in [8]. For the purposes of this article a function G:ω122 will be called measure saturated if it is ν-saturated where ν is the measure on 2 giving each point equal measure.

Lemma 1.5

Assuming the Continuum Hypothesis, there is a symmetric function from ω12 to ω that is category saturated and ν-saturated for every atomic probability measure ν.

Indeed, using the Continuum Hypothesis it is easy to construct G:ω12ω such that Sη(G)=ωη for each ηω1. Note, however, that adding a real will destroy this stronger property; nevertheless, in certain generic extensions the weaker properties of being category saturated or ν-saturated may persist.

Since trees will play a central role in the following discussion, it may be worthwhile reviewing some notation and terminology, even though most of this is standard and almost all of the notation used will follow that of Sections 1.1.D and 7.3.D of [9]. By a tree T will be meant a subset Tω<ω=nωωn that is closed under initial segments — in other words, if tT and k|t| then tkT. If T is a tree and tT then T[t] will denote the tree defined byT[t]={sT|st or ts} and succT(t) will denote the set {sT|st and |s|=|t|+1}.

A tree T will be called infinite splitting if |succT(t)|{1,0} for each tT. Define split(T)={tT||succT(t)|=0} and definesplitn(T)={tsplit(T)||{k|t||tksplit(T)}|=n}. If T is infinite splitting then let ΨT:ω<ωsplit(T) be the unique bijection from ω<ω to split(T) preserving the lexicographic ordering. For tω<ω let Tt=T[ΨT(t)] — the reader is warned that this notation differs from [7]. To emphasize that only elements of a tree are being considered, the notation st will be used to indicate that s=tk for some k|t|.

Let {ui}iω enumerate ω<ω in such a way that if k<|ui| then there is ji such that uik=uj. Then for infinite splitting trees T and S the ordering n is defined by TnS if TS and ΨS(uj)=ΨT(uj) for all jn. Define stem(T)=ΨT() and let Tstem denote {tT|stem(T)t}.

Finally, recall that Miller forcing, or rational perfect set forcing, is denoted by PT in [9] and consists of all infinite splitting trees ordered by inclusion. Laver forcing, on the other hand, is denoted by LT in [9] and consists of all infinite splitting trees T such that Tstem=split(T), also ordered by inclusion. In the case of Laver forcing the notion of a front is useful: If TLT then FT is a front if it consists of incomparable elements of T and every maximal branch of T contains an element of F.

For a tree Wω<ω let max(W) denote the maximal elements of W with respect to inclusion, the natural ordering on the tree. As usual, and in contradiction to common sense, a tree with no infinite branches will be called well founded. If F is a front, then there is a well founded tree W such that F=max(W).

Definition 1.6

If f:ωω+1 define Tf to be nωjnf(j) with inclusion as a tree ordering; the special case when f has constant value ω will be denoted Tω. If f:ωω+1 then a function ψ:Tf[ω1]<0 satisfying that ψ(s)ψ(t)= unless s=t will be said to have disjoint range. In later sections additional requirements will be imposed on functions with disjoint range. If G is a filter of subtrees of Tf and ψ has disjoint range defineS(G,ψ)=tGψ(t) noting that when G is sufficiently generic then ⋂G is a branch through Tf. If G is a generic filter of trees over a model V defineS(G)={S(G,ψ)|ψV and ψ has disjoint range}.

It will be shown that in various generic extensions S(G) is a P-ideal and this will be used in conjunction with the P-ideal dichotomy which is the following the statement () from [10].

Definition 1.7

The statement () says that for every P-ideal I on ω1 one of the following two alternatives holds:

  • 1.

    there is an uncountable Aω1 such that [A]0I

  • 2.

    ω1 can be decomposed into countably many sets orthogonal to I.

The applications of the P-ideal dichotomy in [10] and [11] all rely on simply finding an uncountable set all of whose countable subsets belong to a given ideal. The arguments to be presented here rely on a stronger version of the axiom, but one which is, nevertheless, true in the model constructed in [10]. The following theorem is implicit in Lemma 3.1 of [10]; the following argument simply verifies this assertion.

Theorem 1.8 Abraham & Todorcevic

Let I be a P-ideal on ω1 that is generated by a family of 1 countable sets — in particular, this will hold if 20=1 and I[ω1]0. Then there is a proper partial order PI, that adds no reals, even when iterated with countable support, such that there is a PI-name Z˙ for a subset of ω1 such that for any Yω1 which is not the union of countably many sets orthogonal to I1PIZ˙Y1PI(ηω1)Z˙ηI.

Proof

The proof is the same as that in [10]. To begin, given a non-principal P-ideal I on ω1, fix Aξ such that

  • Aξξ

  • AξI

  • if ξη then AξAη

  • every member of I is almost included in some Aξ.

The partial order PI is defined to consist of pairs p=(xp,Xp) such that:
  • xpI (The reader should not be confused by the claim in [10] that xp can be any countable subset of ω1.)

  • |Xp|0

  • Xp[ω1]1.

The ordering on PI is defined by defining pq if:
  • xpsup(xq)=xq

  • XpXq

  • {ξX|xpxqAξ}Xp for every XXq

and Z˙ is the name for pGxp where GPI is generic. Lemma 3.1 of [10] establishes that if ω1 cannot be decomposed into countably many sets orthogonal to I, then for each γω1 the set of pPI such that xpγ is a dense subset of PI. It will now be verified that the same argument establishes that if Yω1 is not the union of countably many sets orthogonal to I then D(Y)={pPI|xpY} is a dense subset of PI.

To see this suppose that there is no member of D(Y) extending p. Then for each μY there is some XXp such thatX(μ)={ξX|μAξ} is countable. For XXp letB(X)={μY||X(μ)|=0} and note that the hypothesis on Y implies that XXpB(X)Y. It therefore suffices to show that each B(X) is orthogonal to I. To see that this is so, suppose that b is an infinite subset of B(X) and that bI. Then {ξX|bAξ} is clearly a co-countable subset of X because of the cofinality of {Aξ}ξω1. Moreover{ξX|bAξ}=F[b]<0{ξX|(bF)Aξ} and hence there is some F[b]<0 such that {ξX|(bF)Aξ} is uncountable. Then if μbF this contradicts that X(μ){ξX|(bF)Aξ} and |X(μ)|=0 because μB(X). 

Section snippets

Category saturated graphs are universal after adding Miller reals

Lemma 2.1

If TPTS˙S(G˙)andh˙:S˙2 then there is TT and f:ω12 such thatTPTfS˙=h˙.

Proof

Given TPT find T¯T and ψ with disjoint range such that T¯PTS˙=S(G˙,ψ). Now construct Tn and fn such that:

  • 1.

    T0=T¯

  • 2.

    Tn+1nTn

  • 3.

    the domain of fn is jnsΨTn(uj)ψ(ΨTn(s))

  • 4.

    if jn then TnujPTfnsΨTn(uj)ψ(ΨTn(s))f˙.

Then if T=nωTn it is clear that if fnωfn then ψ and f witness that T satisfies the lemma.

To complete the induction it suffices to note that TnunPTsΨTn(uj)ψ(ΨTn(s))S˙ and that

Measure saturated graphs are universal after adding Laver reals

Many of the ideas developed in §2 an be extended to the context of the Laver model. One exception will be Lemma 3.7 that requires the following definition.

Definition 3.1

If ψ has disjoint range and {|ψ(sn)|}nω is bounded for each sTω then ψ will be said to have bounded disjoint range. The definition of S(G,ψ) for ψ with bounded disjoint range is the same as in Definition 1.6, however, if G is a generic filter of trees over a model V defineSb(G)={S(G,ψ)|ψV and ψ has bounded disjoint range}.

Lemma 3.2

If S˙ is a LT

Measure saturated graphs are universal after adding ωω-bounding reals

This section contains the key consistency result needed to establish the main result of this article, namely that the existence of a universal graph on ω1 does not entail the existence of a universal function from ω12 to ω. The key concepts are already contained in §3, the only difference being that the partial order PTf,g of Definition 7.3.3 of [9] will be used in the place of LT. As can be seen in the exposition of PTf,g in [9], there are a great many analogies between this partial order and

The general framework

The results of the preceding sections were originally obtained by a more complicated argument that was eventually replaced by the simpler arguments described in §2, §3 and §4. However, there is one result for which the simplified argument does not seem to be sufficient; this is the question of finding a universal function from ω12 to ω rather than a universal function from ω12 to 2. This section will describe an argument showing that it is consistent with b<d — indeed, non(M)=1<d=2 — that

The key combinatorial lemma

This section describes the combinatorial reason why the existence of a universal graph does not imply the existence of a universal function with range ω.

Lemma 6.1

If b=1 and there is a sequence of pairs of natural numbers {(mi,ni)}iω such that mi<ni<mi+1 for each iω and(F[iω[ni]mi]1)(giωni)(fF)(mω)(km)g(k)f(k) then there is no universal c:ω12ω.

Proof

Let Bη:ηω be a bijection for each ηω1. Suppose that c:ω12ω is a universal function. If ηξω1 and jω letfη,ξ(j)={c(Bη1(k),ξ)nj|kmj}

Allowing embeddings to permute the range

There are various ways of generalizing the notions of embeddings discussed in the introduction. The following two are singled out because something can be said about them.

Definition 7.1

A function U:ω12λ will be called (ρ0,ρ1)-weakly universal if for every f:ω12λ there exists a one-to-one function h:ω1ω1 and functions e0:λλ and e1:λλ such that(αω1)(βω1)e0(f(α,β))=e1(U(h(α),h(β)))if ρi>1 then (ξλ)|ei1(ξ)|<ρiif ρi=1 then (ξλ)ei(ξ)=ξ The triple (h,e0,e1) will be called a (ρ0,ρ1)-weak embedding

Set valued universality

This section will consider weakening the notion of universality for functions by attaching a set of potential values to each edge of a graph. The following definition provides the main idea.

Definition 8.1

Given a family AP(λ) function U:ω12A is A-weakly universal if for every f:ω12λ there exists a one-to-one function h:ω1ω1 such that f(α,β)U(h(α),h(β)) for all α and β in ω1. The function h will be called and A-embedding of f.

Definition 8.2

Consider the following assertions about the existence of certain types of A

Consistency of Kω=k+1 for arbitrary k

All the results up to this point have only established that Kω can consistently take on the values 0 and ω. The goal of this section is to correct this and prove the statement of the title, namely that Kω can have any intermediate value. This will be Theorem 9.1. The proof of Theorem 9.1 is ultimately based on arguments in [1] and [3]. However, while some familiarity with those articles may be useful, the main argument will be presented in detail here since various modifications will need.

A [ω]<0-weakly universal function without any (2,2)-universal function

Note that Theorem 9.1 for the case k=1 yields a model of set theory in which there is no universal function from ω12 to ω yet there is a [ω]2-weakly universal function. The main theorem of this section provides a companion result by showing that there is a model of set theory in which there is an [ω]2-weakly universal function but there is no (a,b)-universal function for any finite a and b. The proof will require some modifications of §9. To begin, Definition 9.5 needs to be augmented.

Definition 10.1

Assume

Open questions

Question 11.1

Is K=k consistent for all k?

Question 11.2

For which a, b, c and d in N does the existence of an (a,b)-universal function imply the existence of an (c,d)-universal function?

Question 11.3

Does the existence of a (1,2)-weakly universal function imply that there is a universal function? See Corollary 7.7.

References (18)

  • S. Shelah

    On universal graphs without instances of CH

    Ann. Pure Appl. Log.

    (1984)
  • A. Dow

    More set-theory for topologists

    Topol. Appl.

    (1995)
  • S. Shelah

    Universal graphs without instances of CH: revisited

    Isr. J. Math.

    (1990)
  • A.H. Mekler

    Universal structures in power 1

    J. Symb. Log.

    (1990)
  • W. Sierpiński

    Sur une fonction universelle de deux variables réelles

    Bull. Int. Acad. Polon. Sci. A

    (1936)
  • R. Rado

    Universal graphs and universal functions

    Acta Arith.

    (1964)
  • P.B. Larson et al.

    Universal functions

    Fundam. Math.

    (2014)
  • W. Chen-Mertens et al.

    Strong colorings over partitions

There are more references available in the full text version of this article.
1

The first author's research for this paper was partially supported by the United States-Israel Binational Science Foundation (grant No. 1838/19), and by the National Science Foundation (grant No. DMS 1833363). This is article has been assigned number 1088 in the first author's list of publications that can be found at http://shelah.logic.at.

2

The second author's research for this paper was partially supported by NSERC of Canada.

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