Halin’s Infinite Ray Theorems: Complexity and Reverse Mathematics: Version E^††thanks: All the authors were partially supported by NSF Grant DMS-1161175.

In this paper we analyze the complexity of several results in infinite graph theory. These theorems are said to be ones of Halin type or, more generally, of ubiquity theory. The classical example is a theorem of Halin [11]: If a countable graph $G$ contains, for each $n$, a sequence $\left\langle R_{0},\ldots,R_{n-1}\right\rangle$ of disjoint rays (a ray is a sequence $\left\langle x_{i}\mid i\in N\right\rangle$ of distinct vertices such that there is an edge between each $x_{i}$ and $x_{i+1}$) then it contains an infinite such sequence of rays. (Note: As will be described in Definition 3.2, when we talk about disjoint rays we always mean pairwise disjoint.) Halin actually deals with arbitrary graphs and formulates the result differently. The uncountable cases, however, are essentially just counting arguments. We deal only with countable structures but discuss his formulation in §6. This standard formulation of his theorem seems like a typical compactness theorem going from arbitrarily large finite collections of objects to an infinite collection. The archetypical example here is König’s Lemma: If a finitely branching tree has paths of length $n$ for every $n$ then it has a branch, i.e. an infinite path. In outline, a modern proof of Halin’s theorem for countable graphs (due to Andreae, see [5, Theorem 8.2.5(i)]) seems much like that of König’s Lemma (and many others in infinite graph theory). The construction of the desired sequence of rays proceeds by a recursion through the natural numbers in which each step is a simple procedure. While the procedure is much more delicate than for König’s Lemma, it is basically of the same complexity. It uses Menger’s theorem for finite graphs at each step but this represents a computable procedure (for finite graphs). The other parts of the step depend on the same type of information as in König’s Lemma. They ask, for example, if various sets (computable in the given graph) are nonempty or infinite. Nonetheless, we prove that the complexity of this construction and theorem are much higher than that for König’s Lemma or other applications of compactness. The concepts from graph theory, computability theory and proof theory/reverse mathematics that we need for our analysis are discussed in §3. Basic references for terminology, background and standard results not explicitly stated or otherwise attributed are Diestel [5] for graph theory; Rogers [22] and Sacks [24] for computability theory; and for reverse mathematics Simpson [28] with an approach which is primarily in terms of formal systems and Hirschfeldt [13] with one primarily emphasizing computability.

We follow two well established procedures for measuring the complexity of constructions and theorems. The first is basically computability theoretic. It has its formal beginnings in the 1950s but has much earlier roots in constructive or computable mathematics reaching back to antiquity. (See Ershov et al. [6] for history and surveys of the approach in several areas of combinatorics, algebra and analysis.) The measuring rod here is relative computability. We say a set $A$ of natural numbers is (Turing) computable from a set $B$, $A\leq_{T}B$, if there is an algorithm (say on a Turing machine or any other reasonable model of general computation) that, when given access to all membership facts about $B$ (an oracle for $B$) computes membership in $A$. The standard hierarchies of complexity here are based on iterations of the Turing jump. This operator takes $B$ to $B^{\prime}$, the halting problem relativized to $B$, i.e. the set of programs with oracle for $B$, $\Phi_{e}^{B}$, such that $\Phi_{e}^{B}$ halts on input $e$. For example, if the tree of König’s Lemma is computable in $B$ then there is a branch computable in the double jump $B^{\prime\prime}$ of $B$.

The second approach is proof theoretic. It measures the complexity of a theorem by the logical strength of the axioms needed to prove it. This approach also has a long history but the formal subject, now called reverse mathematics, starts with H. Friedman’s work in the 1970s (e.g. [7, 8]). One compares axiomatic systems $S$ and $T$ by saying that $T$ is stronger than $S$, $T\vdash S$ ($T$ proves $S$) if one can prove every sentence $\Theta\in S$ from the axioms of $T$. Of course, we know what it means for $\Theta$ to be provable in $S$. The goal here is to characterize to the extent possible the axioms needed to prove a given mathematical theorem $\Theta$. To this end, one begins with a weak base theory. Then one wants to find a system $S$ such that not only does $S\vdash\Theta$ but also $\Theta$ (with the weak base theory) proves all the axioms of $S$. Hence the name reverse mathematics as we seek to prove the “axioms” of $S$ from the theorem $\Theta$. Typically, the systems here are formalized in arithmetic with quantification over sets as well as numbers. The standard base theory ($\mathsf{RCA}_{0}$) corresponds to the axioms needed to do computable constructions. Stronger systems are then usually generated by adding comprehension axioms which assert the existence of specific families of sets. For example, a very important system is $\mathsf{ACA}_{0}$. It is equivalent in the sense of reverse mathematics just described to König’s Lemma. Formally, it asserts that every subset of the natural numbers defined by a formula that quantifies only over numbers (and not sets) exists. This is also equivalent to asserting that for every set $B$, the set $B^{\prime}$ exists.

The early decades of reverse mathematics were marked by a large variety of results characterizing a wide array of theorems and constructions as being one of five specific levels of complexity including $\mathsf{RCA}_{0}$ and $\mathsf{ACA}_{0}$. Each of these systems (Simpson’s “big five”) have corresponding specific recursion theoretic construction principles. In more recent decades, there has been a proliferation of results placing theorems and constructions outside the big five. Sometimes these are inserted linearly and sometimes with incomparabilities. They are now collectively often called the “zoo” of reverse mathematics. (See https://rmzoo.math.uconn.edu/diagrams/ for pictures.)

Theorems and constructions in combinatorics in general, and graph theory in particular, have been a rich source of such denizens of this zoo. Almost all of them have fallen below $\mathsf{ACA}_{0}$ (König’s Lemma) and so have the objects they seek constructible computably in finitely many iterations of the Turing jump. Ramsey theory, in particular, has provided a very large class of constructions and theorems of distinct complexity. One example of the infinite version of a classical theorem of finite graph theory that is computationally and reverse mathematically strictly stronger than $\mathsf{ACA}_{0}$ is König’s Duality Theorem ($\mathsf{KDT}$) for countable graphs. (Every bipartite graph has a matching and a cover consisting of one vertex from each edge of the matching.) The proofs of this theorem for infinite graphs (Podewski and Steffens [21] for countable and Aharoni [1] for arbitrary ones) are not just technically difficult but explicitly used both transfinite recursions and well orderings of all subsets of the given graph. These techniques lie far beyond $\mathsf{ACA}_{0}$. Aharoni, Magidor and Shore [2] proved that this theorem is of great computational strength in that there are computable graphs for which the required matching and cover compute all the iterations of the Turing jump through all computable well-orderings. They also showed that it was strong reverse mathematically as it implied $\mathsf{ATR}_{0}$, the standard system above $\mathsf{ACA}_{0}$ used to deal with such transfinite recursions. Some of the lemmas used in each of the then known proofs were shown to be equivalent to the next and final of the big five systems, $\Pi_{1}^{1}$-$\mathsf{CA}_{0}$ and of corresponding computational strength. Simpson [27] later provided a new proof of the theorem using logical methods that avoided these lemmas and showed that the theorem itself is equivalent to $\mathsf{ATR}_{0}$ and so strictly weaker than the lemmas both computationally and in terms of reverse mathematics.

The situation for the theorems of Halin type that we study here is quite different. The standard proofs do not seem to use such strong methods. Nonetheless, as we mentioned above, the theorems are much stronger than $\mathsf{ACA}_{0}$ with some versions not even provable in $\mathsf{ATR}_{0}$. We prove that these theorems occupy a few houses in the area of the reverse mathematics zoo devoted to what are called theorems (or theories) of hyperarithmetic analysis, THAs (Definition 3.13). Computationally, for each computable well ordering $\alpha$, there is a computable instance of any THA which has all of its required objects Turing above $0^{(\alpha)}$, the $\alpha$th iteration of the Turing jump. On the other hand, they are computationally and proof theoretically much weaker than $\mathsf{ATR}_{0}$ and so $\mathsf{KDT}$. The point here is that there is a single computable graph such that the matching and cover required by $\mathsf{KDT}$ lies above $0^{(\alpha)}$ for all the computable well-orderings $\alpha$, while for each computable instance of a THA there is a computable well-ordering $\alpha$ such that $0^{(\alpha)}$ computes the desired object. In our cases, the instances are graphs with arbitrarily many disjoint rays and the desired object is an infinite sequence of disjoint rays. (The general usage of terms like instances and solutions of a theorem or principle is described at the end of §3.)

Beginning with work of Kreisel [15], H. Friedman [9], Steel [29] and others in the 1960s and 1970s and continuing into the last decade (by Montalbán [16, 17], Neeman [19, 20] and others), several axiomatic systems and logical theorems were found to be THAs and proven to lie in a number of distinct classes in terms of proof theoretic complexity. Until now, however, there has been only one mathematical but not logical example, i.e. one not mentioning classes of first order formulas or their syntactic complexity. This was a result (INDEC) about indecomposability of linear orderings in Jullien’s thesis [14] (see Rosenstein [23, Lemma 10.3]). It was shown to be a THA by Montalbán [16].

The natural quest then became to find out if there are any other THAs in the standard mathematical literature. The issue was raised explicitly in Montalbán’s “Open Questions in Reverse Mathematics” [18, Q30]. As our answer, we provide many examples. Most of them are provable in a well known system above $\mathsf{ACA}_{0}$ gotten by adding on a weak form of the axiom of choice ($\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$).

Several of the basic Halin type theorems (the $\mathsf{IRT}_{\mathrm{XYZ}}$ defined after Definition 3.4) have versions (the $\mathsf{IRT}_{\mathrm{XYZ}}^{\ast}$ of Definition 6.1) like those appearing in the original papers that show that there are always families of disjoint rays of maximal cardinality which are of the same computational strength as the basic versions (Proposition 6.3 and Corollary 6.4). On the other hand, the $\mathsf{IRT}_{\mathrm{XYZ}}^{\ast}$ are strictly stronger proof theoretically than the $\mathsf{IRT}_{\mathrm{XYZ}}$ because they imply more induction than is available in $\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ (Theorem 6.8 and Corollary 6.9). Two of the variations we consider are as yet open problems of graph theory ([3] and Bowler, personal communication). We show that if we restrict the class of graphs to directed forests the principles are not only provable but reverse mathematically equivalent to $\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}+\mathsf{I}\Sigma_{1}^{1}$. Note that as $\mathsf{ATR}_{0}\nvdash\mathsf{I}\Sigma_{1}^{1}$ [28, IX.4.7], these theorems are not provable even in $\mathsf{ATR}_{0}$ or from $\mathsf{KDT}$ (Corollary 6.14). We do not know of other mathematical but nonlogical theorems of this strength. Other versions that require maximal sets of rays (Definition 6.17) are much stronger and, in fact, equivalent to $\Pi_{1}^{1}$-$\mathsf{CA}_{0}$ (Theorem 6.18).

Section 3 discusses basic concepts and background information. The first subsection (3.1) provides what we need from graph theory. Almost all the definitions are standard. At times we give slight variations that are equivalent to the standard ones but make dealing with the computability and proof theoretic analysis easier. We also state the theorems of Halin and some variants that are the main targets of our analysis.

The second subsection (3.2), assumes an intuitive view of computability of functions $f:N\to N$ such as having an algorithm given by a program in any standard computer language (possibly with access to an “oracle” providing information about a given set or function). It then gives the standard notions and theorems that can be found in basic texts on computability theory needed to follow our analysis of the computational complexity of the graph theoretic theorems we study. In particular, it notes the Turing jump operator and its iterations along countable well orderings. These are our primary computational measuring rods. The final subsection (3.3) provides the syntax and semantics for the formal systems of arithmetic that are used to measure proof theoretic complexity. It also describes the standard basic axiomatic systems and their connections to the computational measures of the previous subsection. It includes the formal definition of the class of theorems which includes most of our graph theoretic examples, the THAs, Theorems (or Theories) of Hyperarithmetic Analysis. These are defined in terms of the transfinite iterations of the Turing jump and the hyperarithmetic sets of the previous subsection. In addition it defines $\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ a weak version of the axiom of choice that is an early well known example of such theories and plays a crucial role in our analysis.

Section 4 provides the proof that Halin’s original theorem $\mathsf{IRT}$ (Definition 3.4) is computationally very complicated. For example, given any iteration $0^{(\alpha)}$ of the Turing jump, there is a computable graph satisfying the hypotheses of $\mathsf{IRT}$ such that any instance of its conclusion computes $0^{(\alpha)}$. Indeed, $\mathsf{IRT}$ is a THA. At times, theorems or lemmas are stated in terms of the formal systems of §3.3, but the proofs rely only on the computational notions of §3.2.

Section 5 studies several variations $\mathsf{IRT}_{\mathrm{XYZ}}$ of Halin’s $\mathsf{IRT}$ where we consider directed as well as undirected graphs, edge rather than vertex disjointness for the rays and double as well as single rays. (See Definitions 3.1 and 3.2 and the discussion after Definition 3.4.) We provide reductions over $\mathsf{RCA}_{0}$ between many of the pairs of the eight possible variants. The proofs of these reductions proceed purely combinatorially by providing one computational process that takes an instance of some $\mathsf{IRT}_{\mathrm{XYZ}}$, i.e. a graph satisfying its hypotheses, and produces a graph satisfying the hypotheses of another $\mathsf{IRT}_{\mathrm{X^{\prime}Y^{\prime}Z^{\prime}}}$ and another computable process that takes any solution to the $\mathsf{IRT}_{\mathrm{X^{\prime}Y^{\prime}Z^{\prime}}}$ instance, i.e. any sequence of rays satisfying the conclusion of $\mathsf{IRT}_{\mathrm{X^{\prime}Y^{\prime}Z^{\prime}}}$, and produces a solution to the original instance of $\mathsf{IRT}_{\mathrm{XYZ}}$. (See Propositions 5.3, 5.5 and 5.7 and the associated Lemmas. An additional reduction using a stronger base theory is given in the next section (Theorem 6.15).)

We then show that five of the eight possible variants of $\mathsf{IRT}_{\mathrm{}}$ are THAs (Theorem 5.1). As mentioned in §1, of the remaining three, two are still open problems in graph theory. We do, however, have an analysis of their restrictions to special classes of graphs in Theorem 5.16 and §6.1. The last of the variations, $\mathsf{IRT}_{\mathrm{UED}}$, has been proven more recently by Bowler, Carmesin, Pott [3] using more sophisticated methods than the other results. We have some lower bounds (Theorem 5.9) but we have yet to fully analyze the complexity of their construction.

In the next section (§6) we study some variations of $\mathsf{IRT}$ that ask for different types of maximality for the solutions. The first sort actually follow the original formulation of $\mathsf{IRT}$ in Halin [11]: In any graph there is a set of disjoint rays of maximum cardinality. For uncountable graphs this amounts to a basic counting argument on uncountable cardinals as all rays are countable. When restricted to countable graphs these variations, $\mathsf{IRT}_{\mathrm{XYZ}}^{\ast}$, are easily seen to be equivalent to our more modern formulation by induction. Technically, the induction used is for $\Sigma_{1}^{1}$ formulas ($\mathsf{I}\Sigma_{1}^{1}$) which is not available in $\mathsf{RCA}_{0}$. More specifically we show (Proposition 6.3) that $\mathsf{IRT}_{\mathrm{XYZ}}+\mathsf{I}\Sigma_{1}^{1}$ and $\mathsf{IRT}_{\mathrm{XYZ}}^{\ast}+\mathsf{I}\Sigma_{1}^{1}$ are equivalent (over $\mathsf{RCA}_{0}$). As the definition of THAs only depends on standard models where full induction holds, if $\mathsf{IRT}_{\mathrm{XYZ}}$ is a THA then so is $\mathsf{IRT}_{\mathrm{XYZ}}^{\ast}$.

We then prove that these maximal cardinality variants $\mathsf{IRT}^{\ast}_{\mathrm{XYZ}}$ are strictly stronger proof theoretically than the basic $\mathsf{IRT}_{\mathrm{XYZ}}$ (when they are known to be provable in $\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$). This is done by showing (Theorem 6.8 and the Remark that follows it) that the relevant $\mathsf{IRT}^{\ast}_{\mathrm{XYZ}}$ all imply weaker versions of $\mathsf{I}\Sigma_{1}^{1}$ that are analogous to the restrictions of $\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$ embodied in weak (or unique)-$\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$ and finite-$\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$ (Definitions 7.1 and 7.2). In all the cases, it is enough induction to prove (with the apparatus of the basic $\mathsf{IRT}_{\mathrm{XYZ}}$) the consistency of $\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$ and so by Gödel’s second incompleteness theorem they cannot be proved in $\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$ (Corollary 6.9).

As for proving full $\Sigma_{1}^{1}$ induction from an $\mathsf{IRT}_{\mathrm{XYZ}}^{\ast}$ we are in much the same situation mentioned above for $\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ and $\mathsf{IRT}_{\mathrm{XYZ}}$. In particular, $\mathsf{IRT}_{\mathrm{DVD}}^{\ast}$ and $\mathsf{IRT}_{\mathrm{DED}}^{\ast}$ for directed forests each proves $\mathsf{I}\Sigma_{1}^{1}$ as well as $\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ (Theorems 6.12 and 6.13) and so are equivalent to $\mathsf{I}\Sigma_{1}^{1}+\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$. As before, this shows that they are strictly stronger than $\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ (Corollary 6.14). Indeed, as mentioned at the end of §1, they are not even provable in $\mathsf{ATR}_{0}$. We do not know of any other mathematical theorem with this level of reverse mathematical strength.

The second variation of maximality, $\mathsf{MIRT}_{\mathrm{XYZ}}$, studied in §6.2 is also mentioned in the original Halin paper [11]. It asks for a set of disjoint rays which is maximal in the sense of set containment. Of course, this follows immediately from Zorn’s Lemma for all graphs. For countable graphs we provide a reverse mathematical analysis, showing that each of the $\mathsf{MIRT}_{\mathrm{XYZ}}$ is equivalent to $\Pi^{1}_{1}$-$\mathsf{CA}_{0}$ (Theorem 6.18).

In §7, we discuss the reverse mathematical relationships between the THAs associated with variations of Halin’s theorem and previously studied THAs as well as one new logical THA (finite-$\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ of Definition 7.1). Basically, all the $\mathsf{IRT}_{\mathrm{XYZ}}^{\ast}$ (and so $\mathsf{IRT}_{\mathrm{XYZ}}+\mathsf{I}\Sigma_{1}^{1}$) imply H. Friedman’s $\mathsf{ABW}_{0}$ (Definition 7.4) by Theorem 7.7 and finite-$\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ (Theorem 7.3). On the other hand, none of them are implied by it (Theorem 7.10) or by $\Delta_{1}^{1}$-$\mathsf{CA}_{0}$ (Definition 7.8 and Theorem 7.9). $\mathsf{ABW}_{0}+\mathsf{I}\Sigma_{1}^{1}$ does, however, imply finite-$\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ which is not implied by weak (unique)-$\Sigma_{1}^{1}\text{-}\mathsf{AC}_{0}$ (Goh [10]). Figure 4 summarizes many of the known relations with references.

In the penultimate section (§8) we study the only use of $\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$ in each of our proofs of $\mathsf{IRT}_{\mathrm{XYZ}}$. It consists of $\mathsf{SCR}_{\mathrm{XYZ}}$ which says we can go from the hypothesis that there are arbitrarily many disjoint rays to a sequence $\langle X_{k}\rangle_{k}$ in which each $X_{k}$ is a sequence of $k$ many disjoint rays. We analyze the strength of the $\mathsf{SCR}_{\mathrm{XYZ}}$ and the weakenings $\mathsf{WIRT}_{\mathrm{XYZ}}$ of $\mathsf{IRT}_{\mathrm{XYZ}}$ which each take the existence of such a sequence $\langle X_{k}\rangle_{k}$ as its hypothesis in place of there being arbitrarily many disjoint rays. For example, for all the $\mathsf{IRT}_{\mathrm{XYZ}}$ which are consequences of $\Sigma^{1}_{1}\text{-}\mathsf{AC}_{0}$ and so are THAs, $\mathsf{IRT}_{\mathrm{XYZ}}$ is equivalent to $\mathsf{SCR}_{\mathrm{XYZ}}$ over $\mathsf{RCA}_{0}$ (Corollary 8.5) and so all of them are also THAs. For the same choices of $XYZ$, $\mathsf{ACA}_{0}$ proves $\mathsf{WIRT}_{\mathrm{XYZ}}$ over $\mathsf{RCA}_{0}$. While a natural strengthening of $\mathsf{WIRT}_{\mathrm{XYZ}}$ does imply $\mathsf{ACA}_{0}$ and indeed is equivalent to it (Theorem 8.9), we do not know if $\mathsf{WIRT}_{\mathrm{XYZ}}$ itself implies $\mathsf{ACA}_{0}$. All we can prove is that it is not a consequence of $\mathsf{RCA}_{0}$ (Theorem 8.10).

In the last section (§9), we mention some open problems.

We begin with basic notions and terminology from graph theory. At times we use formalizations that are clearly equivalent to more standard ones but are easier to work with computationally or proof-theoretically. The following two subsections supply background and basic information about the standard computational and logical/proof theoretic notions that we use here to measure the complexity of the graph theorems and constructions that we analyze in the rest of this paper. Note that we denote the set of natural numbers by $N$ when we may be thinking of them in a model of arithmetic as in §3.3 and by $\mathbb{N}$ when we emphasize that we specifically want the standard natural numbers.

We devote this section to the proof of

In this section we consider only vertex-disjoint single rays in undirected graphs, as in the statement of $\mathsf{IRT}$. The proof of Theorem 4.1 will be split into two parts. The first part verifies that $\mathsf{IRT}$ satisfies the second clause of Definition 3.13.

It follows that $\mathsf{IRT}$ is not provable in $\mathsf{ACA}_{0}$, despite the apparent similarity between $\mathsf{IRT}$ and a compactness result. (Indeed, $\mathsf{IRT}_{\mathrm{}}$ is not even provable in $\Delta_{1}^{1}$-$\mathsf{CA}_{0}$ (Theorem 7.9).)

Next, we present essentially the proof of $\mathsf{IRT}$ attributed to Andreae (see [5, Theorem 8.2.5 and bottom of pg. 275]) and then analyze it with an eye to the axioms which can be used to formalize it. We then use this analysis to complete the proof of Theorem 4.1.

The key combinatorial lemma implicit in Andreae’s proof is:

Before proving Lemma 4.3, let us use it to prove $\mathsf{IRT}$.

It remains to prove Lemma 4.3. The key ingredient is Menger’s theorem for finite graphs. If $A$ and $B$ are disjoint sets of vertices in a graph, we say that $P$ is an $A$-$B$ path if $P$ starts with some vertex in $A$ and ends with some vertex in $B$. A set of vertices $S$ separates $A$ and $B$ if any $A$-$B$ path contains at least one vertex in $S$.

We now present the proof of Lemma 4.3.