Representing embeddability as set inclusion

A few steps are made towards representation theory of embeddability among uncountable graphs. A monotone class of graphs is defined by forbidding countable subgraphs, related to the graphs end-structure. Using a combinatorial theorem of Shelah it is proved: The complexity of the class in every regular uncountable A > Mi is at least A+ + sup{/x : jx+ < A} For all regular uncountable A > Mi there are 2 pairwise non embeddable graphs in the class having strong homogeneity properties. It is characterized when some invariants of a graph G G Q\ have to be inherited by one of fewer than A subgraphs whose union covers G. All three results are obtained as corollaries of a representation theorem (Theorem 1.10 below), that asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality A > Mi onto set inclusion over all subsets of reals or cardinality A or less. Continuity properties of the homomorphism are used to extend the first result to all singular cardinals below the first cardinal fixed point of second order. The first result shows that, unlike what Shelah showed in the class of all graphs, the relations of embeddability in this class is not independent of negations of the GCH. M. Kojman: embeddability §0 Introduction The study of embeddability among infinite structures has a long tradition of invoking combinatorics. One well known example is Laver's use of Nash-Williams' combinatorial results to show that embeddability among countable order types is well quasi ordered [L]. In the study of embeddability among uncountable structures, the most prominent combinatorial principle has been the Generalized Continuum Hypothesis (GCH), which asserts that every infinite set has the least possible number of subsets. Hausdorff proved as early as 1914 using the GCH that in every infinite power there is a universal linear ordering, that is, one in which every linear ordering is embedded as a subordering. Jonsson [Jo] used the GCH to prove that classes of structures satisfying a list of 6 axioms have universal structures in all uncountable powers. See also [R] for graph theory and [MV] for model theory. Finer combinatorial principles have come from Jensen's work in Godel's universe of constructible sets [Je]. Thus, for example, Macintyre [M] uses Jensen's diamond — a principle stronger than CH — to prove that no abelian locally finite group of size Ni is embeddable in all universal locally finite groups of size Hi, and Komjath and Pach [KP1] use the same principles to prove that there is no universal graph in power Ni among all graphs omitting Ku^x. A common property of the combinatorial principles mentioned above is that they are not provable from the usual axioms of Set Theory. Easton [E] showed that the GCH can fail for all regular cardinal. Magidor [Ma] showed the GCH could fail at N^ with GCH below it and Foreman and Woodin [FW] showed that the GCH could fail everywhere (both using large cardinals). In Spite of this, the common impression among mathematicians working in areas having intimate relations to infinite cardinals, like infinite graph theory, infinite abelian groups, and model theory, remained that the GCH was a useful assumption, while its negations were not. In the context of embeddability this impression was fortified by Shelah's independence results. Shelah showed that universal structures in uncountable powers may or may not exist under negations of the GCH. Thus, while GCH implies the existence of universal graphs in all infinite A, the assumption A < 2° for regular uncountable A does not determine the 1 u r . , < . . . „ . . . M. Kojman: embeddability existence or non existence of a universal graph in A (see [S3], [Me] and [K]). Shelah's independence results [S 1,2,3] created the expectation that the existence of a universal structure in a class of structures in uncountable cardinalities would always be independent of negations of GCH, unless the existence was trivial (because the class of structures is "dull"). See, for example, [KS] for results about the class of lif^-free graphs that support this expectation. The understanding of negations of the GCH at singular cardinals has changed dramatically in the last five years. The most fascinating development in this area is Shelah's bound on the exponents of singular cardinals. Shelah proved the following magnificent theorem, formulated here, though, in a way Shelah himself resents: 0.1 Theorem: If 2 < #„ for all n then 2*» < NW4. Knowing that by Cohen's results no bound can be put on the exponent of a regular cardinal, this theorem is exceptionally thrilling. A short proof of it can be found in [J]. The formulation Shelah prefers is the following: 0.1a Theorem: cf ([NW] °,C) < N^ This formulations says that the cofinality of the partial ordering of set inclusion over countable subsets of K^ is ALWAYS smaller than N^, no matter how large N °̂ may be. In other words, this theorem exposes a robust structure of the partial ordering of set inclusion, which is affected by negations of the GCH in a limited way only. The reader will verify that 0.1a implies 0.1. A proof of this theorem is in Shelah's recent book on Cardinal arithmetic [S]. In this book Shelah reduces the problem of computing the exponent of a singular cardinal to an algebra of reduced products of regular cardinal, and uses a host of new and sophisticated combinatorics to analyze the structure of such reduced products. A common property of the combinatorial principles Shelah uses in [S] and in later works on cardinal arithmetic, is that they are proved in ZFC, the usual axiomatic framework of set theory. This is necessary, since 0.1a (unlike the conclusion of 0.1) is an absolute theorem, namely assumes nothing about cardinal arithmetic. In this paper we use some of Shelah's combinatorics to expose robust connections between the structure of embeddability over a monotone class of infinite graphs and the M. Kojman: embeddability relation of set inclusion. This is done by means of a representation theorem, that asserts the existence of a surjective homomorphism from the former relation onto the latter. One corollary is that the structure of embeddability over the class we shall study — which is defined by imposing restrictions on the the graph's end-structure — is not independent of negations of GCH, but also information that is not related to cardinal arithmetic is obtained. Shelah's ZFC combinatorics on uncountable cardinals was found useful in the study of embeddability in several papers. In [KjSl] it was shown that if A > Ni is regular and A < 2**° then there is no universal linear ordering in A. In other words, an appropriate negation of CH determines negatively the problem of existence of a universal linear ordering in power A. Similar results were proved for models of first order theories [KjS2]; infinite abelian groups [KjS3] and [S4] and metric spaces [S5]. But so far no application was found for infinite graphs, in spite of the existing rich and active theory of universal graphs. The theory of universal graphs, that began with Rado's construction [R] of a countable strongly universal graph, has advanced considerably since, especially in studying universality over monotone classes (see [DHV] for motivation for this). A monotone class of graphs is always of the form Forb (F), the class of all graphs omitting a some class F of "forbidden" configurations as subgraphs. A good source for the development of this theory is the survey paper [KP1], in which the authors suggest a generalization of universality, which they name "complexity": the complexity of a class of graphs is the least number of members in the class needed to embed as induced subgraphs all members in the class. The complexity is 1 exactly when a strongly universal graphs exists in the class. The paper is organized as follows. In Section 1 a class of graphs is specified by forbidding countable configurations related to the graph's end-structure, and it is noted that by a generalization of a theorem by Diestel, Halin and Vogler the complexity of the resulting class Q at power A is at least A"". A surjective homomorphism is now constructed from the relation of (weak) embeddability over Q\ for regular A > N2 onto the relation of setinclusion over all subsets of reals of cardinality < A. Combining both results, max{ A+, 2**° } is set as a lower bound for the complexity of Gx for regular A > Ni. In Section 2 a certain continuity property of the homomorphism from Section 1 is M. Kojman: embeddability proved, and is used to extend the lower bound from Section 1 to all singular cardinals below the first fixed point of second order. In this Section the representation Theorem is stated in its full generality, generalizing Theorem 1.8 to higher cardinals. In Section 3 it is proved that in every regular A > Ni there are 2 pairwise non mutually embeddable elements in Q\, each of which being "small" in the sense that it is mapped by the homomorphism to a finite set. For the case A inaccessible, this result makes use of a very recent result by Gitik and Shelah about non-saturation of the non-stationary ideal on A. No cardinal arithmetic assumptions are made in this Section and in Section 4. In Section 4 a decomposition theorem is proved for a proper subclass of Q\, A > Ni regular, which is also defined by forbidding countable configurations. The Theorem gives a necessary and sufficient condition to when the invariant of a graph G in the class is inherited by at least one subgraphs from a collection of < A subgraphs whose union covers G. NOTATION A graph G is a pair (V, E) where V is the set of vertices and E C [V] is the set of edges. By G[v] we denote the neighbourhood of v G V in G, namely {u eV : {v, u] G E}. A graph G is bipartite if there is a partition G = G\ U (?2 of G to two (non-empty) disjoint independent vertex sets, each of which is called a side. An ordinal is a set which is well ordered by G. A cardinal is an initial ordinal number. The cardinality | A| of a set A is the unique cardinal equinumerous with A. The cofinality cf A of a cardinal A is the least cardinal K such that A can be represented as a union of K sets, each of cardinality less than A. A cardinal A is singular if cf A < A and is regular if cf A = A. If K, K! are cardinals we denote by KK the complete graphs on K vertices and by KKik f the complete bipartite graphs with K vertices in one side and K' in the other. If G\ is isomorphic to a subgraph of G2 we write G\ the complexity of Gx, be the least cardinality of a subset D C Gx with the property that for every G G Gx there exists G' G D such that G —) * a Partially ordered set. Let covA\, the covering number of [A], be the least cardinality of a subset D C [A] with the property that for every B G [A] there exists B' G D such that B C B'. We remark that the least cardinality of a dominating subset is defined for every quasiordered set, and bears the name "cofinality"; but we stick here to the customary graphtheoretic and set theoretic existing terminologies and refer to the former as "complexity" and to the latter as "covering number". 0.2 Definition: Let A be an uncountable regular cardinal. A club of A is a closed (in the order topology) and unbounded subset of A. Club sets generate a filter over A, indeed a X-complete filter: the intersection of fewer than A subsets of A, each of which contains a club, contains a club. A subset of A is called stationary if its intersection with every club of A is non empty. The ideal of all subsets of A which are disjoint to some club of A is the non-stationary ideal. Thus club sets are analogous to measure 1 sets, non-stationary sets are measure zero and stationary sets are positive measure (meet every measure 1 set). Let S be {a : a < A A cf a = K} and SQ = {a : a < A A cf a = LO}. We shall need the following combinatorial tool: M. Kojman: embeddability 0.3 Theorem: (Shelah) If A > is regular, /x a cardinal and fi < A then there is a stationary set S C A and a sequence C = (c$ : 5 e S) with otp cs = ^ and sup c* = 5 such that for every dosed unbounded E C X the set N(E) := {5 e S : c$ C E} is stationary. For a proof see I[Sh-e, new VI§2]= [Sh-e, old III§7]. A sequence C as in the theorem is called a "club guessing sequence". If the cs are thought of as "guesses", then the theorem says that for every club (measure 1) set stationarily many (positive measure) of the guesses are successful. Suppose C is a club guessing sequence as above. We define two guessing ideals over A, id(C) and id(C), as follows: 0-4 Definition: (0) X G id(C) iff for some club E C A it holds that cs%E for all 5 G S fl E. (1) X G id(C) iff for some club E C A it holds that cs %* E for all S G S D E, where cs ^* means that an end segment of cs is contained in E. Thus a set X C A is in id(C) iff there are no stationarily many 5 G X such that cs is contained in E for some club E, and X C A is in \&{C) iff there are no stationarily many S G X such that cs is almost (=except for a proper initial segment) contained in E for some club E. The ideal id(C) is a A-complete ideal over A and id (C) is normal. Also, id (C) C id(C). 0.5 Definition: Let u be the set of natural numbers. Let Fin be the set of all finite subsets of co. Two subsets I , 7 C w are equivalent mod Fin iff the symmetric difference X \Y UY \ X G Fin. By V(u>) we denote the power set of w and by P(u) we denote V(u)/Fin the set of all equivalence classes of subsets of u> modulo Fin. Finally, we need a few definitions about reduced powers. A reduced power is a generalization of ultra-power. 0.6 Definition: Suppose that A is a structure, A a cardinal and F a filter over A. Let A be the set of all functions from A to the structure A and let A/F be the reduced power of A modulo F. M. Kojman: embeddability §1 Representing embeddability as set inclusion In [K] it is proved: 1.1 Theorem: If Q is a class of graphs that contains all K^^-free incidence graphs of A C V(OJ) and the cofinaJity of the continuum is Ni, then cpQ\ > 2° for all uncountable A<2*°. In particular, if cf 2° = Ni there is no universal graph (in the class of all graphs) in any uncountable A < 2°. On the other hand, Shelah proved in [S3]: 1.2 Theorem: If A is regular uncountable, it is consistent that A < 2° and that a universal graph in power A exists. Mekler [Me] generalized Shelah's result to more general classes of structures. Both result together can be understood as follows: a singular 2° affects the structure of embeddability in a broad spectrum of classes of infinite graphs below the continuum, but a large regular 2° may have no effect on the class of all graphs and the classes handled by Mekler and Shelah. It is reasonable to ask if for some "reasonably defined" class of graphs for which the structure of embeddabilty below 2° is influenced by the size of 2°. In this section we show that forbidding certain countable configurations gives rise to a class with such a desired connection. The configurations we forbid are related to the end structure of graphs. 1.3 Definition: A ray in a graph G is a 1-way infinite path. A tail of a ray R C G is an infinite connected subgraph of R. Two rays in G are tail-equivalent iff they share a common tail. Tail-equivalence is an equivalence relation on rays. We mention in passing that tail-equivalence is a refinement of end-equivalence. For more on both relations see [D]. 1.4 Definition: Let Q be the class of all graphs G satisfying that for every v € G the induced subgraph of G spanned by G[v] has at most one ray up to tail-equivalence. 1.5 Claim: There is a non-empty set T of countable graphs, each containing an infinite path, such that Q = Forb (F). M. Kojman: embeddability Proof. : Let F' be the set of all countable graphs that contain (at least) two tail-inequivalent rays and let F be all graphs obtained by choosing an element from F' and joining a new vertex to all its vertices. If a graph G contains a subgraph in F then G fi Q. Conversely, Suppose that G g Q. Let v G G be a vertex such that there are two rays RuR.2 Q G[v] which are not tail-equivalent. Let G' C G be the induced subgraph spanned by {v} U J?i U R2. Now G' G F and so G £ Forb (F). Graphs with forbidden countable configuration that contain an infinite path were considered by Diestel, Halin and Vogler in [DHV] for A = NoThey prove (Theorem 4.1): 1.6 Theorem: (Diestel-Halin-Vogler) Let T be a non-empty set of countable graphs each containing an infinite path. Then Qx0 = (7NO(F) has no universal element This Theorem applies to our class Q by 1.5 above, and because every forbidden configuration in F contains an infinite path. The following is a straightforward generalization of Theorem 1.6, and is included only for completeness of presentation's sake: 1.7 Theorem: Let T be a non-empty set of countable graphs each containing an infinite path. Then wcpForb A(F) > A for all infinite cardinals A. Proof. By induction on a < A define graphs Ga as follows: Ga is obtained by joining a vertex wa to a disjoint union of Gp for all /3 < a (such that wa is not in this union). For all A < a < A the graph Ga contains no infinite path and therefore belongs to Forb A(r). Suppose that T is a collection of A graphs and / a : Ga -» G(a) is an embedding of Ga into some graph G(a) e T. By the pigeon hole principle there is a fixed graph G G T such that G = G(a) for A many a < A"". A second use of the pigeon hole principle gives a vertex w(0) G G and an unbounded set I C A + such that w(0) = / a (^a) for all a £ X. This implies, by the construction of the Ga 's, that G[w(0)] contains as subgraphs copies of Ga for unboundedly many a < A + and therefore of all a < A. Repeating this argument a set {w(n) : n < tu} C G is found that spans in G a copy of Ku. Therefore G contains all countable configurations and therefore does not belong to Forb \(F). A Thus for every infinite A we have w c p ^ > A. Also Theorem 1.1 from the previous section applies to G, because Q contains all bipartite graphs; thus (setting 6 = No), 2 if cf 2 < A (we use here wcp < cp). M. Kojman: embeddability The virtue of G is, nevertheless, that wcpGx > max{A,2°} regardless to the cardinality of the continuum for all regular A > Ni (and many singular A, as seen in the next section). This is a corollary of the following: 1.8 Theorem: If A > Ni is regular then there is a surjective homomorphism 4> : {Gx, Ni is regular then wcpGx > max{A+,2*}. This theorem will be extended to singular values of A in the next section. We turn now to the proof of the theorem. The homomorphism $ will be factored through a reduced product of the inclusion relation over subsets of reals. We will prove the following stronger formulation: 1.10 Theorem: Suppose that A > Ni is regular. Then (0) there is a surjective homomorphism $ : (Gx, ([R-]~i Q) (1) $ from (1) can be chosen to be a composition i/jcp where ip is a surjective homomorphism to a reduced power ([R]-, C) / / for some normal ideal I over A. Proof. : First let us notice that ([R]-,C) is a homomorphic image of ([R]-,C) / / for every ideal /: Suppose that A is a representative of an equivalence class of ([R]-)/I. Define t/>([A]) := {x G R : {S < A : x G A(<5)} g I}. In words, rl>([A]) is the set of all reals that appear in a positive set of coordinates. It is routine to check that the definition of I/J does not depend on the choice of a representative and that ip is a homomorphism. Thus it suffices to prove that there is a surjective homomorphism ([R]-, C) / / for some normal ideal / over A. This is in fact more than needed for (0). The set R can be replaced here by any set of equal cardinality. It is convenient for us to work with P(UJ) = V(u)/Fin. We shall define a mapping tp : (Gx,<) -> ([^(^)]~ ,Q)// after specifying / . We shall show that <p is well defined, is a homomorphism and is surjective. For the definition of the mapping we fix a club guessing sequence ~C = (cs : 5 G S), S C A stationary and