Hypergraphs and proper forcing
Given a Polish space and a countable collection of analytic hypergraphs on , I consider the -ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.
The purpose of this paper is to introduce a class of proper forcing notions which can be in a natural way associated with hypergraphs on Polish spaces. There is a number of theorems which connect simple combinatorial properties of the hypergraphs with deep forcing properties of the resulting forcing notions. The story begins with the concept of analytic hypergraphs and their associated -ideals:
Let be a Polish space.
A hypergraph on is a subset of ;
if is a hypergraph on , a -anticlique is a set such that ;
if is a countable collection of hypergraphs on , then is the -ideal on -generated by the Borel subsets of which happen to be anticliques in at least one of the hypergraphs in . -ideals generated by countable collection of analytic hypergraphs are called hypergraphable.
In the spirit of , I will be interested in the quotient posets of Borel sets positive with respect to the ideal , ordered by inclusion. Such posets may fail to be proper already in very simple circumstances. Still, a great majority of definable, proper forcings preserving Baire category in the literature can be conveniently presented as quotient forcings of hypergraphable ideals. They can be so presented, but invariably they are not, since the authors have not been aware of the existence and the great advantages and comfort of such a presentation. The purpose of this paper is to change this unfortunate situation.
In Section 2 I isolate two several broad classes of hypergraphs for which the quotient is proper: the actionable hypergraphs, associated with a countable group action on the underlying Polish space and the nearly open hypergraphs. Theorem 2.2 shows the properness of the actionable posets; these in fact can be canonically decomposed into (as opposed to regularly embedded to) a two step -closed*c.c.c. iteration–Theorem 2.11. The nearly open hypergraphs give posets with quite different properties, one of the distictions being that they generate a minimal forcing extension unless they add a Cohen real.
Section 3 shows that there is an enormous amount of general information that one can derive just from the very basic properties of the generating hypergraphs. For example, if the generating hypergraphs all have arity two, then in the resulting extension, every element of is a branch through some ground model -branching tree on –Corollary 3.7. If each of the generating hypergraphs has finite arity, then the poset has the Sacks property–Corollary 3.12. If the edges of the hypergraphs can be diagonalized in a natural sense, then the resulting forcing is bounding–Corollary 3.26. If the hypergraphs have a simple Fubini property, then the quotient forcing preserves outer Lebesgue measure–Corollary 3.36. In any case, the hypergraphable forcings preserve the Baire category–Corollary 3.43. There are useful combinatorial criteria for dealing with compact anticliques in closed graphs, such as the ones presented in Theorem 3.47 or 3.59. Other such criteria concern adding an independent real, Corollary 3.57 or Theorem 3.62. Each of these theorems is adorned with a number of examples, some new, others well-known. Other similar theorems will appear in forthcoming work.
It may seem that one could classify quotient posets coming from very restrictive hypergraph classes. Section 4 attempts to do just that for some types of invariant graphs. There are two well-known examples, the Silver forcing (Example 2.7) and the Vitali forcing (Example 2.8), and all the other posets in the category are in a precise sense between these two. Many invariant graphs give rise to quotient posets which are naturally isomorphic to one of these two–Theorems 4.5 and 4.6. However, there are some intermediate examples such as the Vitali-odd forcing (Example 4.10), and many examples where I cannot determine what exactly is happening, typically connected with fine additive or group combinatorics (Subsection 4.4). Subsection 4.5 shows that there is a natural proper poset derived from the (non-invariant) KST graph and proves an attendant dichotomy.
Section 5 studies operations on hypergraphs that lead to operations on partial orders. The countable support product is studied in Section 5.2, and it turns out that in the case of posets given by actionable families of finitary hypergraphs, the product is hypergraphable again and the computation of the associated ideal is so simple that it gives rise to many preservation properties for product that seem to be very awkward to obtain in any other way. One can also represent countable support iterations, even illfounded ones, using natural operations on hypergraphs, Subsection 5.3.
The last section of the paper provides the descriptive complexity computations necessary to push all the proofs through. The computations were mostly known previously, either as folklore theorems or as published results.
There is an overwhelming number of open questions; I will mention two of strategic nature. The classes of analytic hypergraps isolated in this paper do not exhaust the class of hypergraphable proper forcings by any stretch. The most natural question in the area is wide open:
Characterize the analytic hypergraphs such that the quotient poset is proper.
It is just as unclear to me which proper partial orders can be presented as hypergraphable. It is clear from the work in this paper that such a presentation advances the understanding of the forcing properties of the poset more than any other piece of information. The only two significant restrictions on the class of hypergraphable forcing are that the associated -ideal is on and the poset preserves Baire category. However, within these limitations there are many posets which I suspect cannot be presented as hypergraphable, such as the product of two copies of Sacks forcing. Thus, another question begs an answer:
Characterize the idealized forcings on Polish spaces which can be presented as hypergraphable.
The notation follows the set theoretic standard of . If is a -ideal on a Polish space, the symbol denotes the poset of Borel -positive subsets of the space, ordered by inclusion. A hypergraph is finitary if all its edges are finite, possibly of arbitrarily large finite sizes. If is a class of hypergraphs, a -ideal is -hypergraphable if there is a a countable family of analytic hypergraphs in the class such that ; thus, I speak of finitary hypergraphable ideals, graphable ideals, nearly open hypergraphable ideals etc. I will need precise terminology regarding maps between hypergraphs: suppose that are Polish spaces with respective hypergraphs on them, and suppose that is a continuous injection. I will say that a function is a homomorphism of to if for every edge , the sequence belongs to . The function is a reduction of to if for every sequence , . The function is a near reduction if can be decomposed into countably many Borel sets for such that the function is a reduction of to for every number . Note that if is a near reduction of to , then the - image of any Borel -anticlique decomposes into countably many Borel -anticliques and the -preimage of any Borel -anticlique decomposes into countably many Borel -anticliques. Thus, any near reduction of to transports the ideal to restricted to the range of . If and are countable families of hypergraphs on the respective Polish spaces , then a near reduction of to is a continuous injection together with a bijection such that for every , is a near reduction of to . As before, a near reduction clearly transports the ideal to the ideal restricted to the range of .
2 Properness theorems
2.1 Actionable hypergraphs
Let be a Polish space and a countable set of analytic hypergraphs on . Say that is actionable if there is a countable group and a Borel action of on such that
every edge of every hypergraph in is a subset of a single orbit;
for every and every , .
A -ideal on is actionable if there is an actionable collection of hypergraphs on such that is -generated by Borel sets which are anticliques in at least one of the hypergraphs in .
Suppose that is an actionable -ideal on a Polish space . Then the poset of Borel -positive sets ordered by inclusion is proper.
Let be a countable family of analytic hypergraphs and let be a countable group with an action witnessing the assumption that the ideal is actionable. Let be the -name for the generic element of the space . The next claim yields a key homogeneity feature of the poset .
Let be any element. The map induces an automorphism of the ordering .
Note that if a Borel set is a -anticlique for some graph and is any element, then is a Borel -anticlique and . This means that the set of generators of and consequently the whole ideal is invariant under the action of the group. The claim immediately follows. ∎
The second preparatory claim describes a key derivative operation on conditions in the poset .
Let be a Borel -positive set and . Then the set is analytic and -positive.
Suppose that the set belongs to the ideal . Then, since the ideal is by definition generated by Borel sets, there must be a Borel set which is a superset of . The definition of the set shows that the set is a -anticlique and so the set is a Borel -anticlique. By the definitions, the set is in the ideal , contradicting the initial assumptions. ∎
Let be a countable elementary submodel of a large structure. The poset still adds a single point such that the generic filter is the collection of those sets in which contain , and the action still induces automorphisms of the poset . The following claim is central.
If and is a Borel -anticlique, then .
Suppose that is a Borel set and is a condition forcing ; I must find a -edge in . Let ; by Claim 2.4, this is an analytic -positive subset of in the model . By Fact 6.1, it contains a Borel -positive subset . Now, suppose that be a generic filter containing the condition and the associated generic point. Then , and so there is an edge such that is in its range. By the initial assumptions on the hypergraph , the range of the edge is contained in the orbit of , and so each point in the range of is also generic for the poset . Since the points on the range of all belong to the set , by the forcing theorem they must all belong to the set . Thus, the Borel set fails to be a -anticlique in the extension , and by the Mostowski absoluteness it cannot be a -anticlique in the ground model either. ∎
Let is a condition, then the set is -generic over the model is Borel and -positive.
The Borelness of the set of generics over countable models is a general fact, proved in [21, Fact 1.4.8]. To see that , suppose that are Borel anticliques for some hypergraphs in ; I must produce a point . To this end, let be a countable elementary submodel of large structure containing for and as elements, let be a filter generic over the model containing the condition , and let be its associated generic real. By Claim 2.5 applied in the model , , and by the Mostowski absoluteness between the models and , holds as required. ∎
The theorem now immediately follows by the characterization of properness in [21, Proposition 2.2.2]. ∎
Let be the Silver graph on , connecting points just in case they disagree on exactly one entry. The graph is invariant under the usual action of the rational points of the Cantor group on the whole group. The quotient poset is well known to have a dense subset naturally isomorphic to the Silver forcing [20, Theorem 2.3.37].
Let be the Vitali graph on , connecting points just in case they disagree on only finite number of entries. The graph is invariant under the usual action of the rational points of the Cantor group on the whole group. The quotient poset is well-known to have a dense subset naturally isomorphic to the Vitali forcing, or -forcing as it is called in [21, Section 4.7.1]. The main difference between the Vitali and Silver forcings is that Vitali forcing adds no independent reals while Silver forcing does, even though below I identify another profound iterable difference–Corollary 3.49.
Let be the KST graph (for Kechris–Solecki–Todorcevic ) on . To define it, let be binary strings for each such that the set is dense in . Put if differ in exactly one entry and their longest common initial segment belongs to the set . It is well-known and easy to prove that the KST graph is closed, acyclic, and spans the Vitali equivalence relation. Borel anticliques of must be meager. Let be the family of all rational shifts of the graph . By the definitions, the family is actionable. The quotient poset does not depend on the initial choices as proved in Subsection 4.5 and I call it the KST forcing. The main iterable difference between the KST forcing and the Silver forcing is that in the Silver extension, ground model coded compact anticliques of any closed acyclic graph still cover their domain Polish space–Corollary 3.48. In the KST extension this clearly fails for the initial graph .
Let be a countable group acting continuously on a Polish space and let be an invariant probability measure on . The hypergraph consisting of all elements such that the set consists of pairwise orbit equivalent points and has -positive closure is certainly invariant. Does the quotient forcing depend on the initial choice of the action and the measure?
2.2 The canonical intermediate extension
The generic extensions associated by the actionable posets share a certain important feature: there is a large, natural intermediate forcing extension. The main theorem of this section identifies the most important features of this extension.
Let be a Polish space and an actionable -ideal on it; let be a generic filter. Then there is an intermediate extension such that a set of ordinals belongs to just in case its intersection with every ground model countable set belongs to the ground model. Moreover, is a c.c.c. extension of .
The second sentence identifies the intermediate model uniquely; a moment’s thought will show that must be the largest, in the sense of inclusion, intermediate model of ZFC with no new reals. The last sentence implies that is a nontrivial extension of the ground model except in the case that is equivalent to the Cohen forcing: if then is c.c.c. below some condition, and the only definable c.c.c. poset preserving Baire category is the Cohen forcing by . A good part of the theorem is the precise identification of the intermediate model ; the details of this have been incorporated into the proof rather than the statement of the theorem.
Start in the ground model . Let be a family of analytic hypergraphs generating the -ideal , with the associated action of a countable group on . Write for the resulting orbit equivalence relation on ; is Borel and all its classes are countable. Mover to the generic extension and write for the generic point added by the filter . The model is the class of all sets in which such that every element of the transitive closure of is in definable from parameters in and the additional parameter . It is well-known that classes of this type are models of ZFC. The following claim provides the central piece of information about the model .
Move to the ground model and let be a condition, a -name, and ; I will find a condition and a point such that . Strengthening the condition , I may identify the parameters in and the formula which defines in with those parameters plus the parameter . Strengthening the condition even further, I can find a Borel function such that . I will find a Borel -positive set such that is constant on -classes, and then I argue that for every such a function there must be an -positive Borel set on which is constant. Clearly, if is the constant value, then as required.
For the construction of the set , let be a countable elementary submodel of a large structure containing all the objects named so far. Let be the set of all points -generic over the model ; by Theorem 2.2, this set is Borel and -positive. To see that the function is constant on -classes, suppose that are -related points and is a group element such that . Both points are -generic over and so (identifying with its transitive collapse) I may consider the generic extensions , . Since , these two generic extensions coincide. By the forcing theorem, the formula applied to the parameters and the -class containing both gives a point which must be equal to both and . Thus as desired.
For the construction of the set , suppose for contradiction that for every , the set belongs to the ideal . Then the set of all pairs such that has all vertical sections in the ideal . By Fact 6.1, there are Borel sets and hypergraphs for each such that and each vertical section of the set is a -anticlique. By the -additivity of the -ideal , there is a number such that the Borel set is -positive. Find a -edge in the set . Since the edge is a subset of a single -class, the function is constant on the edge with value , then a contradiction with the assumption that the section is -anticlique appears. ∎
It immediately follows that if is a set of ordinals such that for some countable set , , then : if were an element of , so would be the set , contradicting the claim. On the other hand, suppose that is a set of ordinals such that for every countable set . Let be a -name in the ground model such that . In the model , the set is countable and definable from and the equivalence class . There is a countable set of ordinals such that whenever are distinct sets of ordinals then they disagree on the membership of some ordinal in . Since is a proper extension of , there is a countable set of ordinals in the ground model such that . By the assumption on the set , the intersection belongs to the ground model, and so can be defined from as the only set of ordinals in whose intersection with is equal to . Thus, and the second sentence of the theorem is proved.
To evaluate the properties of the forcing that leads from to , one has to evaluate the forcing that leads from to . This is a result of an entirely general procedure. Step back into the ground model. Let be the poset of all -positive Borel -invariant sets, ordered by inclusion.
is a filter on generic over .
Move to the ground model. To prove the genericity, suppose that is a condition and is a Borel -invariant -positive set. It will be enough to find a condition in such that its -saturation is a subset of . That way, it will be confirmed that the map is a projection of to , and the genericity follows. ∎
Now, the remainder poset leading from the model to is the poset of all Borel -positive subsets coded in the ground model such that . The following is easy to show via a genericity argument.
Let be a point -related to . The collection of all sets containing is a filter generic over . Moreover, .
It follows that . On one hand, the filter is definable in from the parameter as the set of all -invariant Borel sets coded in the ground model which contain as a subset. This shows that . To prove the opposite inclusion, move to the model . Suppose that is an -name for a set of ordinals. Suppose that is a condition forcing that is definable in the -extension using a formula with parameters in and an additional parameter . It will be enough to show that the condition decides the membership of all ordinals in , since then as desired. To see this, note that the filters for in the claim cover the whole poset and since their generic points are all -related, the filters all evaluate the formula and so the name in the same way.
It also follows that is c.c.c. since the poset is covered by countably many filters in the extension which has the same as . The theorem has just been proved. ∎
2.3 Nearly open hypergraphs
There is an entirely different class of hypergraphable proper forcings which is perhaps more frequent in the literature, even though never in the following most convenient presentation:
Let be an analytic hypergraph on a Polish space . The hypergraph is nearly open if for each edge and every there are open sets containing such that for every choice of points , the sequence is an edge in the hypergraph .
It is clear from the definition that the first vertex in an edge of a nearly open hypergraph may carry priviledged information about the edge, and in the more interesting examples this is indeed the case. Note that I do not demand the hypergraphs to be invariant under the permutation of vertices in the edges. There is an important subclass of the nearly open hypergraphs, namely the hypergraphs which are open in the box topology on . In these hypergraphs no vertex in an edge is clearly priviledged, and the treatment becomes much simpler; in particular, the associated -ideal is -generated by closed sets.
Unlike the case of actionable ideals, the definition of a nearly open hypergraph depends on the choice of the topology on the underlying space , and it may occur that one needs to make a rather unnatural change of topology to present a given hypergraph as nearly open. Note that the poset and all its properties do not depend on the topology of as long as the algebra of Borel sets remains the same.
Let be a family of nearly open hypergraphs on a Polish space . The quotient poset is proper.
Write . Let be a countable elementary submodel of a large structure and consider the (countable) poset . As in the case of actionable ideals, the following claim will be central.
Let be a hypergraph and be a Borel -anticlique. Then .
Suppose on the contrary that some condition forces . Shrinking the set if necessary, I may assume that for every open set , or . The set is a coanalytic -anticlique. Its complement cannot belong to , since then it would be covered by a Borel set and would be the union of and the -anticlique and therefore in . Now, by Theorem 6.1(2), the -positive set has a Borel -positive subset . Now, in choose a sufficiently generic filter on the poset containing and let be its associated generic point. By the forcing theorem, it will be the case that .
Now, since , there is an edge such that and . Use the openness of the graph to find basic open sets for such that and for every choice of points , the sequence is an edge in the hypergraph . For every , the set is nonempty, containing , and so it is -positive. It also belongs to the model . Thus, I can choose a sufficiently generic filter on the poset containing the set and let be its generic point. By the forcing theorem, holds.
Finally, consider the tuple . It is an edge in the hypergraph , and it consists of points in the set . This contradicts the assumption that was a -anticlique. ∎
Let be a condition in the model . I must show that the set is -generic over the model is -positive. To this end, let for be Borel subsets of which are anticliques in at least one hypergraph on the generating family . I must produce a point which is -generic over and does not belong to the set . To do this, choose a sufficiently generic filter on the poset containing the set , and let be its generic point. Clearly, , and by the claim as desired. The proof is complete! ∎
If is a on -ideal on a Polish space generated by closed sets, then there is a nearly open and in fact box open hypergraph on such that . Just let just in case the closure of does not belong to .
Let be Polish spaces and be a Borel function. Let be the hypergraph on consisting of those tuples such that but there is an open set containing and no other points of . The hypergraph is not nearly open as it stands, but becomes nearly open if the topology of the space is updated to make the function continuous. The -ideal is generated by Borel subsets of on which the function is continuous (with the original topology on the space ).
The -ideals generated by nearly open graphs differ from the actionable ideals in many significant ways. One remarkable difference is Theorem 3.51 proved below, which provides for many iterable preservation properties of the quotient forcings of the nearly open hypergraphs that the actionable ideals can never have.
In the only result in this subsection, I isolate a rather humble class of hypergraphable posets which are not proper.
Let be a locally countable, acyclic, analytic graph on a Polish space and let be the -ideal -generated by Borel -anticliques. The quotient forcing is either trivial or not proper.
Let be the equivalence relation connecting points if there is a -path from to . By the second reflection theorem, is a subset of a countable Borel equivalence relation . By the Feldman–Moore theorem, there is a Borel action of some countable group which induces as the orbit equivalence relation. Consider the following a small claim of independent interest which does not use the acyclicity assumption:
If is a Borel -positive set, then there is a Borel -positive set such that two elements of are connected by a -path if and only if they are connected by a -path whose vertices are all in .
Since the Borel chromatic number of on is uncountable, by a result of Miller  there is a continuous map which is a homomorphism of the KST graph to and antihomomorphism of the Vitali equivalence relation to . Since the KST graph spans the Vitali equivalence relation, the compact set is a witness to the validity of the claim. ∎
Now, let be a countable elementary submodel of a large structure containing , and the action. By the properness criterion [21, Proposition 2.2.2], it will be enough to show that the set is -generic over is a -anticlique. Suppose towards contradiction that there are points which are -connected. Let be an element such that . By the forcing theorem applied in the model , there must be a Borel set which forces to be -generic and -connected to . Now, thinning down the set successively, I may arrange that for each , , , and finally, by the claim, that two elements of are connected by a -path if and only if they are connected by a -path whose vertices are all in .
Look at the Borel set . Since is -generic, the Borel set cannot be a -anticlique. Thus, there are points such that . Thus, is a -path from to . By the choice of the set , there must be also such a path using exclusively points from the set . The two paths together form a cycle in the graph , which is a contradiction. ∎
3 Fubini-type preservation properties
In this section, I provide several results which, from simple combinatorial properties of the collection of hypergraphs , obtain central forcing properties of the quotient posets. The theorems are proved through the Fubini property with various -ideals generated by Suslin ergodic forcings. This approach is in an important class of cases optimal (Theorem 3.45) and has the advantage of automatically yielding preservation theorems for countable support iteration and product (Corollary 5.8 and 5.26). Recall the central definition:
[21, Definition 3.2.1] Let be -ideals on respective Polish spaces . The ideals have the Fubini property () if for every Borel -positive set , every Borel -positive set , and every Borel set , either has a -positive vertical section or the complement of has a horizontal -positive section.
The Fubini property is always going to be verified through the following property of Suslin partial orders and the attendant theorem:
Let be an analytic family of hypergraphs on a Polish space . Let be a Suslin forcing. The notation denotes the following statement: for every Borel -positive set , every Borel function , and every hypergraph , there is a condition which forces that there is an edge which consists of ground model elements of the set , and is a subset of the generic filter.
One good way to satisfy the statement is to actually find an edge such that the set has a lower bound in the poset , which then will serve as the condition . This is clearly the only way to act if the edges in the hypergraph are finite. However, in the case of hypergraphs of infinite arity, this approach may not be flexible enough, and the sought edge may be found only in the -extension.
The following theorem is one of the reasons why hypergraphable forcings are so special. To state it, if is a forcing, is a Polish space and is a -name for an element of , write for the -ideal generated by the analytic sets such that .
Let be an analytic family of hypergraphs on a Polish space such that the quotient forcing is proper. Let be a Suslin c.c.c. forcing, a Polish space, a -name for an element of . Then, implies .
Suppose that holds. To prove the theorem, suppose that and are Borel and -positive sets respectively and is a Borel set with all vertical sections in the ideal . I must prove that the complement has an -positive horizontal section.
Suppose for contradiction that this fails. Then, by Theorem 6.1(3), there are Borel sets and hypergraphs for such that and each vertical section of is a -anticlique. By Theorem 6.14, there is a Borel -positive set , a natural number , and a Borel function such that for all , the condition forces .
Now, use the assumption to find a condition and a -name such that , consists of ground model elements of , and for each , belongs to the generic filter. Let be a countable elementary submodel of a large structure containing all objects named so far, and let be a filter -generic over , and let and . By the forcing theorem, satisfies and . By the Mostowski absoluteness, these statements transfer from to and so is a -edge consisting of points in the horizontal section . This contradicts the choice of the set . ∎
3.1 The localization property
The quotient posets arising from families of analytic hypergraphs in which every edge is finite with a fixed bound on its arity share a strong preservation property.
A Suslin poset is analytic --linked if it can be covered by the union of countably many analytic sets such that if are conditions in one of the analytic sets, then they have a common lower bound.
Suppose that is a number and is a countable family of analytic hypergraphs on a Polish space such that each hypergraph in has arity . Suppose that is a Suslin poset which is analytic --linked. Then .
Let be a countable cover of the poset by analytic -linked pieces. Suppose is a Borel -positive set, is a Borel function, and is a hypergraph. For each number , the preimage is analytic. If it is a -anticlique, by the first reflection theorem it can be extended to a Borel -anticlique . If this occurred for each number , I would have , contradicting the assumption that . Thus, there is a number and an edge such that . Since the set is -linked and the set has size at most , there is a lower bound of . This lower bound witnesses the statement . ∎
Among the corollaries of the theorem, one appears to be exceptionally powerful:
 Let be a natural number. A poset has the -localization property if every element of in the -extension is a branch through some -branching tree in the ground model.
Suppose that is a number and is a countable family of analytic hypergraphs on a Polish space such that each hypergraph in has arity . If the poset is proper, then it has the -localization property.
I will need a poset adding a large -branching tree. Let be the partial order of all pairs where is a finite -branching tree, is a finite set, every element of contains a terminal node of as an initial segment, and every terminal node of is an initial segment of at most one element of . The ordering is defined by if is an end-extension of and . The poset is Suslin c.c.c., and in fact any many conditions sharing the same first coordinate have a lower bound. This also proves that the poset is analytic --linked.
The poset adds a generic -branching tree which is the union of the first coordinates of conditions in the generic filter. A simple genericity argument shows that every ground model element has a finite modification which is a branch through . Let be a the space of all -branching trees on , and let be the -ideal generated by those analytic sets such that .
Suppose that is a condition and is a -name for an element of . By the Borel reading of names in proper forcing, it is possible to thin out the condition to find a Borel function such that . Let be the set of all pairs such that no finite modificaton of is a branch through the tree . The vertical sections of belong to the -ideal . By Theorems 3.5 and 3.3, there has to be an -branching tree such that the horizontal section of the complement of the set associated with , the Borel set a finite modification of is a branch through the tree is -positive. The condition forces some finite modification of to be a branch through , and so to be a branch to some finite modification of the tree . ∎
For a fixed number consider the hypergraph of arity on the space which contains exactly all tuples such that for some , while all functions are identical on the set . The resulting poset is the wider version of the Silver forcing. It does have the -localization property, while it fails to have the -localization property.
Let be the graph on connecting points if the smallest such that is even. Let be the graph on connecting points if the smallest such that is odd, and let . The resulting forcing is the forcing studied by Kojman among others [21, Section 4.1.5], . Since the two graphs are open, the poset is proper, and by Corollary 3.7 it has the -localization property.
3.2 The Sacks property
If one drops the demand that there be a uniform bound on the arities of edges in all hypergraphs in the generating collection, while each hypergraphs contains only edges of finite and bounded arity, then the resulting poset still maintains strong preservation properties.
Suppose that is a countable family of analytic hypergraphs on a Polish space such that for each there is such that has arity . and is a hypergraphable -ideal on . Suppose that is a Suslin poset which is analytic --centered for every . Then .
Suppose is a Borel -positive set, is a Borel function, and is a hypergraph. Let be the arity of , and let be a countable cover of the poset by analytic -linked pieces. For each number , the preimage is analytic. If it is a -anticlique, by the first reflection theorem it can be extended to a Borel -anticlique . If this occurred for each number , I would have , contradicting the assumption that . Thus, there is a number and an edge such that . Since the set is -linked and the set has size at most , there is a lower bound of . This lower bound witnesses the statement . ∎
Among the corollaries of the theorem, one again appears to be dominant. The requisite definitions:
An narrow tunnel is a function such that for all , . A point is enclosed by the tunnel if for all , . A poset has the Sacks property if every point in in the -extension enclosed by a ground model narrow tunnel.
All the hypergraphable posets in the class identified in Theorem 3.10 have the Sacks property.
I will need a Suslin poset which adds a generic narrow tunnel. Let be the set of all pairs such that is a function with domain such that for all , is a set of size . Also, is a set of size . The ordering is defined by if , , and for all and all , holds. It is not difficult to see that the forcing is c.c.c. and in fact analytic -centered for every .
The forcing adds a generic narrow tunnel which is the union of the first coordinates of all conditions in the generic filter. A genericity argument shows that every ground model element of has a finite modification which is enclosed by the generic tunnel. Let be a the space of all narrow tunnels, and let be the -ideal generated by those analytic sets such that .
Suppose that is a condition and is a -name for an element of . By the Borel reading of names in proper forcing, it is possible to thin out the condition to find a Borel function such that . Let be the set of all pairs such that no finite modification of is not enclosed by the tunnel . The vertical sections of belong to the -ideal . By Theorems 3.10 and 3.3, there has to be a narrow tunnel such that the horizontal section of the complement of the set associated with , the Borel set a finite modification of is enclosed by is -positive. The condition forces some finite modification of to be enclosed by , and so to be enclosed by some finite modification of the tunnel . ∎
Let and be the family of analytic hypergraphs on in which a -edge is a set such that for some interval of length , the sequences in are equal off while every function in is a subset of some element of . Then is actionable as witnessed by the standard group action. The associated poset is the poset of all partial functions from to whose codomain contains arbitrarily long intervals, ordered by reverse inclusion.
Note that in the previous example there is no number such that all hypergraphs in the generating family have arity . Thus, the conclusion is that the poset has the Sacks property by Corollary 3.12, but does not have the -localization property for any number . To see the failure of the -localization property, consider the function in the generic extension which to each assigns the binary string such that for each , the value of the generic point at is equal to .
3.3 The weak Sacks property
If one further loosen the demands on the hypergraphs by allowing each of the hypergraphs to contain edges of all finite arities, the Sacks property may fail. Instead, I obtain the following.
Let be a Suslin poset. is analytic -centered if it can be written as a union of countably many analytic centered pieces.
Suppose that is a countable family of analytic hypergraphs on a Polish space such that all of the edges of all hypergraphs in are finite. Suppose that is a Suslin poset which is analytic -centered. Then holds.
Let be a countable cover of the poset by analytic centered pieces. Suppose is a Borel -positive set, is a Borel function, and is a hypergraph. For each number , the preimage is analytic. If it is a -anticlique, by the first reflection theorem it can be extended to a Borel -anticlique . If this occurred for each number , I would have , contradicting the assumption that . Thus, there is a number and an edge such that . Since the set is centered and the set is finite, there is a lower bound of . This lower bound witnesses the statement . ∎
Again, one of the consequences of the theorem seems to be dominant. For the definitions,
A point is enclosed by a narrow tunnel on an infinite set if for all , . A poset has the weak Sacks property if every point in in the -extension enclosed by a ground model narrow tunnel on a ground model infinite set.
The hypergraphable posets in the class of Theorem 3.15 have the weak Sacks property.
I will need a Suslin poset which adds a generic narrow tunnel and an infinite set. Let be the set of all triples such that is a function with domain such that for all , is a set of size . Also and is a finite set. The ordering is defined by if , , and for all and all , holds.It is not difficult to see that the forcing is analytic -centered; in fact, any finite set of conditions with the same first and third coordinate has a lower bound.
The forcing adds a generic narrow tunnel which is the union of the first coordinates of all conditions in the generic filter, and a generic infinite set which is the union of the third coordinates of the conditions in the generic filter. A genericity argument shows that the every ground model element of has a finite modification which is enclosed bythe tunel on the set . Let be the space of all pairs where is a narrow tunnel and is an infinite set, and let be the -ideal of all analytic set such that .
To prove the corollary, suppose that is a condition and is a -name for an element of . Thinning the condition if necessary, I may find a Borel function such that . Let be the set of all triples such that no finite modification of is enclosed by the tunnel on the set . Then the vertical sections of the Borel set are -small. By Theorems 3.10 and 3.3, there must be an -positive horizontal section of the complement. This horizontal section, call it , is attached to a pair . Clearly, a finite modification of is enclosed by the tunnel on the set , and so is enclosed by the tunnel on a finite modification of the set . ∎
It is easy to see that the weak Sacks property implies the bounding property. Thus, I can conclude
Let be a hypergraphable -ideal on a Polish space . If every edge of every hypergraph in the generating family is finite and the quotient poset is proper, then the quotient poset is bounding.
The following corollary uses an entirely different -centered poset, derived from the classical Solovay coding.
Let be a hypergraphable -ideal on a Polish space such that every edge of every hypergraph in the generating family is finite and the quotient poset is proper. In the -extension, for every infinite set there is a ground model coded Borel set such that both sets , are infinite.
I need the requisite Suslin poset. Fix a Borel function such that the range of consists of pairwise almost disjoint infinite subsets of . The poset can be viewed as an iteration of adding a Cohen generic function from to and then coding the resulting partition of into a Borel set. More specifically, a condition is a pair such that is a finite function from to and is a finite set. The ordering is defined by if , , and for all and all such that , .
The poset is clearly Suslin. It is also analytic -centered. To see this, if is finite, is a finite collection of pairwise disjoint basic open subsets of and is a function, consider the set of those conditions such that , and for every and every holds. It is not difficult to see that the set is analytic and centered. Moreover, the poset is covered by the countably many sets obtained in this way.
Let be the -name for the union of the second coordinates of the conditions in the generic filter. A simple genericity argument shows that for every infinite set , the set is infinite is forced to be infinite co-infinite in . Let and let be the -ideal of all analytic sets such that .
To prove the corollary, suppose that is a condition and is an infinite set. Shrinking to a countable set and thinning out if necessary, it is possible to find Borel functions such that is an injective enumeration of the set . Let be the set of those pairs for which the set is infinite is either finite or cofinite. The vertical sections of the set belong to the ideal , and so by Theorems 3.15 and 3.3 there is such that the horizontal section of the complement of corresponding to is -positive. Let is infinite and observe that and are both infinite sets. ∎
Let be a function and let be the closed subset of all functions pointwise . Let be the graph of all tuples such that there is such that the functions agree everywhere except on , and the set contains all numbers . The hypergraph is actionable as witnessed by the group of finite support product of the cyclic groups of size . Corollary 3.18 shows that the quotient poset is bounding. It is the wide version of the Silver forcing introduced in [2, Definition 7.4.11]. Note that the hypergraph has edges of arbitrarily large finite sizes, so Theorem 3.10 does not apply and indeed, the poset does not have the Sacks property as is immediately obvious from looking at its natural generic point.
Another family of hypergraphs with finite edges only, whose quotient poset fails to have the Sacks property, is associated with packing measures and it is discussed in Example 3.41.
3.4 The bounding property
It turns out that there is a useful criterion for the hypergraphs to generate a bounding quotient forcing.
Let be an analytic hypergraph on a Polish space . Call diagonalizable if for every collection of edges in such that the points are identical for all , there are finite sets and an edge such that .
Note that I do not expect the hypergraph to be closed under permutation of vertices in the edges. Thus, can be formulated so that the first vertex