On Homomorphism Graphs

We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for Δ>2 it is impossible to give a simple characterization of acyclic Δ-regular Borel graphs with Borel chromatic number at most Δ: such graphs form a Σ^1_2-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.

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1 Introduction

Descriptive combinatorics is an area concerned with the investigation of combinatorial problems on infinite graphs that satisfy additional regularity properties (see, e.g., [Pik21, KM20] for surveys of the most important results). In recent years, the study of such problems revealed a deep connection to other areas of mathematics and computer science. The most relevant to our study are the connections with the so-called model from the area of distributed computing. There are several recent results that use distributed computing techniques in order to get results either in descriptive combinatorics [Ber20, Ber21, BCG21, Ele18, GR21a], or in the theory of random processes [HSW17, GR21b].

The starting point of our work was the investigation of the opposite direction. Namely, our aim was to adapt the celebrated determinacy technique of Marks [Mar16] to the model of distributed computing. In order to perform the adaptation (which is indeed possible, see our conference paper [BCG21]111The connection between the current paper and the conference paper [BCG21] is the following. The latter paper builds a theory of local problems on trees from several perspectives and aims to a broader audience. The original version of this paper should have been a journal version of some results from [BCG21] aiming to people working in descriptive combinatorics. In the end, we added several new applications of our method that cannot be found in [BCG21].), we had to circumvent several technical hurdles that, rather surprisingly, lead to the main objects that we study in this paper, homomorphism graphs (defined in Section 3). We refer the reader to [BCG21] for a detailed discussion of the concepts and their connections to the model.

Before we state our results, we recall several basic notions and facts. A graph on a set is a symmetric subset of . We will refer to as the vertex set of , in symbols , and to as the edge set. If , an -coloring of is a mapping such that . The chromatic number of , is the minimal for which an -coloring exists. If and are graphs, a homomorphism from to is a mapping that preserves edges. Note that if and only if admits a homomorphism to the complete graph on vertices, . We denote by the supremum of the vertex degrees of . In what follows, we will only consider graphs with degrees bounded by a finite number, unless explicitly stated otherwise. A graph is called -regular if every vertex has degree . It is easy to see that . Moreover, Brooks’ theorem states that this inequality is sharp only in trivial situations: if , it happens if and only if contains a complete graph on vertices, and if , it happens if and only if

contains an odd cycle.

We say that is a Borel graph if is a standard Borel space, see [Kec95], and the set of edges of is a Borel subset of endowed with the product Borel structure. The Borel chromatic number, , of is defined as the minimal for which a Borel -coloring exists, here we endow with the trivial Borel structure. Similar concepts are studied when we relax the notion of Borel measurable to merely measurable

with respect to some probability measure, or

Baire measurable with respect to some compatible Polish topology.

It has been shown by Kechris-Solecki-Todorčević [KST99] that , and it was a long standing open problem, whether Brooks’ theorem has a literal extension to the Borel context, at least in the case . For example, it has been proved by Conley-Marks-Tucker-Drob [CMTD16] that in the measurable or Baire measurable setting the answer is affirmative. Eventually, this problem has been solved by Marks [Mar16], who showed the existence of -regular acyclic Borel graphs with Borel chromatic number . Remarkably, this result relies on Martin’s Borel determinacy theorem, one of the cornerstones of modern descriptive set theory.

Results

First let us give a high-level overview of the strategy (for the precise definition of the notions discussed below see Section 3). Fix a . To a given Borel graph we will associate an acyclic Borel graph of degrees bounded by . Roughly speaking, the vertex set of the graph will be a collection of pairs , where and is a homomorphism from the -regular infinite rooted tree to that maps the root to , and is adjacent to if is obtained from by moving the root to a neighbouring vertex.

The main idea is that the we can use the combinatorial properties of to control the properties of . Most importantly, we will argue that from a Borel -coloring of we can construct a -coloring of : to each we associate games analogous to the ones developed by Marks, in order to select the “largest” set among the sets for , and color with the appropriate . As this selection will be based on the existence of winning strategies, the coloring of will not be Borel. However, it will still be in a class that has all the usual regularity properties (we call this class weakly provably , see Section 2 for the definition of this class and the corresponding chromatic number, ). Thus we will be able to prove the following.

Theorem 1.1.

Let be a locally countable Borel graph. Then we have

In particular, holds if the Ramsey measurable (if ), Baire measurable, or measurable chromatic number of is .

Next we list the applications. In each instance we use a version of Theorem 1.1 for a carefully chosen target graph . These graphs come from well-studied contexts of descriptive combinatorics, namely, Ramsey property and Baire category.

a) Complexity result.

We apply homomorphism graphs in connection to projective complexity and Brooks’ theorem. One might conjecture that the right generalization of Brooks’ theorem to the Borel context is that Marks’ examples serve as the analogues of the complete graph, i.e., whenever is a Borel graph with , then must contain a Borel homomorphic copy of the corresponding example of Marks. Note that in the case this is the situation, as there is a Borel analogue of odd cycles that admits a homomorphism into each Borel graph with (see [CMSV21]).

In [TV21] it has been shown that it is impossible to give a simple characterization of acyclic Borel graphs with Borel chromatic number . The construction there was based on a Ramsey theoretic statement, the Galvin-Prikry theorem [GP73]. An important weakness of that proof is that it uses graphs of finite but unbounded degrees. Using the homomorphism graph combined with the method developed in [TV21] and Marks technique, we obtain the analogous result for bounded degree graphs.

Theorem 1.2.

For each the family of -regular acyclic Borel graphs with Borel chromatic number has no simple characterization, namely, it is -complete.

From this we deduce a strong negative answer to the conjecture described above.

Corollary 1.3.

Brooks’ theorem has no analogue for Borel graphs in the following sense: there is no countable family of Borel graphs such that for any Borel graph with we have if and only if for some the graph contains a Borel homomorphic copy of .

b) Chromatic number and hyperfiniteness.

Recall that a Borel graph is called hyperfinite, if it is the increasing union of Borel graphs with finite connected components. In [CJM20] the authors examine the connection between hyperfiniteness and notions of Borel combinatorics, such as Borel chromatic number and the Lovász Local Lemma. Roughly speaking, they show that hyperfiniteness has no effect on Borel combinatorics, for example, they establish the following.

Theorem 1.4 ([Cjm20]).

There exists a hyperfinite -regular acyclic Borel graph with Borel chromatic number .

Using homomorphism graphs, we provide a new, short and more streamlined proof of this result. In particular, the conclusion about the chromatic number follows from our general result about (a version of ), while to get hyperfiniteness we can basically choose any acyclic hyperfinite graph as a target graph. To get both properties at once, we simply pick a variant of the graph (see [KST99, Section 6]) as our target graph.

c) Graph homomorphism.

We also consider a slightly more general context: homomorphisms to finite graphs. Clearly, the -regular examples constructed by Marks do not admit a Borel homomorphism to finite graphs of chromatic number at most , as this would imply that their Borel chromatic number is . No other examples of such graphs were known. We show the following.

Theorem 1.5.

For every and every there are a finite graph and a -regular acyclic Borel graph such that and does not admit Borel homomorphism to . The graph can be chosen to be hyperfinite.

This theorem is a step towards the better understanding of Problem 8.12 from [KM20].

Roadmap.

The paper is structured as follows. In Section 2 we collect the most important definitions and theorems that are going to be used. Then, in Section 3 we establish the basic properties of homomorphism graphs and their various modifications. Section 4 contains Marks’ technique’s adaptation to our context, while in Section 5 we prove our main results. We conclude the paper with a couple of remarks Section 6.

2 Preliminaries

For standard facts and notations of descriptive set theory not explained here we refer the reader to [Kec95] (see also [Mos09]).

Given a graph , we refer to maps and as vertex ()-labelings and edge ()-labelings, respectively. An edge labeling is called an edge coloring, if incident edges have different labels. Let be a family of subsets of , and . An measurable -coloring is an -coloring of such that for each . Using this notion, we define the measurable chromatic number of , to be the minimal for which such a coloring exists.

We denote by the collection of infinite subsets of the set , and by the family of finite sequences of elements of . Define the shift-graph (on ), , by letting and be adjacent if or . The shift-graph has a close connection to the notion of so called Ramsey property: for finite and with let . A set is called Ramsey if for each set of the form there exists such that or (see, e.g., [KST99, Kho12, Tod10] for results on the shift-graph and Ramsey measurability). The following follows from the definition.

Theorem 2.1.

The graph has no Ramsey measurable finite coloring.

Note that the Galvin-Prikry theorem asserts that Borel sets are Ramsey measurable. However, adapting Marks’ technique to our setting will require the usage of families of sets that are much larger than the collection of Borel sets. By () formulas we mean the collection of second-order formulas over the structure , of the appropriate form (see, e.g., [Jec03, Section 17]). A set is called provably , if there are and formulas and and an such that and it is provable (from ZFC) that (see, [Kan09]).

It will be convenient to consider a slightly more general notion that is sufficiently robust. Assume that is a Polish space and . We say that is weakly provably if there exists a Borel map such that where is provably . We will use the following result.

Proposition 2.2.

Let be a Polish space. Weakly provably subsets of

  1. form an algebra,

  2. have the Baire property,

  3. are measurable with respect to any Borel probability measure,

  4. in the case have the Ramsey property.

Proof.

The first assertion is easy from the definition. In order to see the second and third assertions note that it has been shown by Solovay (unpublished) and Fenstad-Norman [FN74] that provably subsets of are universally measurable and are -universally Baire. Since these families are closed under taking Borel preimages, we are done. In an upcoming paper we will show that weakly provably sets have the Ramsey property [GKV]; the proof also essentially follows from [IS89, Theorem 3.5]. ∎

If is a nonempty pruned tree, and , will denote the two-player infinite game on with legal positions in and payoff set . We will call the first player and the second . Note that the Borel Determinacy Theorem [Mar75] states that one of the players has a winning strategy in whenever is Borel. Recall that a subset of a Polish space is in the class if there is some Borel set such that

Remark 2.3.

To establish Theorem 1.2, Theorem 1.4 and Theorem 1.5, we only need that the sets in the class have the regularity properties listed in Proposition 2.2, which is a weaker statement than the one above. However, we were unable to locate a reference for this fact that avoids the usage of weakly provably -sets. Note also that Proposition 2.2 easily follows from determinacy (i.e., the assumption that in one of the players has a winning strategy if is ).

We will need a coding for Borel sets. Let be a set of Borel codes and sets and with the properties summarized below:

Proposition 2.4.

(see [Mos09, 3.H])

  • , , ,

  • for and we have ,

  • if is a Polish space and then there exists a Borel map so that and for every we have .

Moreover, in the case the sets , , and can be described by , , and formulas, respectively.

Using a recursive encoding we identify trees with elements of . It turns out that sets that arise from the existence of winning strategies in Borel games are weakly provably .

Lemma 2.5.

Let be a Polish space and be Borel. Then the set

is weakly provably .

Proof.

To ease the notation set , and . Note that we have , , . Consider first the set

We show that is provably . Note that is a winning strategy for in if and only if is a strategy for such that . Using Proposition 2.4 this yields that can be described by a formula. Moreover, as whenever , and is Borel, by the Borel determinacy theorem has a winning strategy in if and only if has no winning strategy. That is, for every strategy for we have . This yields an equivalent description of using a formula.

To deduce the lemma from this, applying a Borel isomorphism between and some subset of we may assume that . It follows that the set above is Borel isomorphic to a section of , which shows our claim. ∎

3 The homomorphism graph

In this section we define the main objects of our study, homomorphism graphs, and establish a couple of their properties.

Let be a countable group and be a generating set. Assume that is an action of on the set . As there is no danger of confusion we always denote the action with the symbol . The Schreier graph of such an action is a graph on the set such that are adjacent iff for some we have that .

Probably the most important example of a Schreier graph is the (right) Cayley graph, that comes from the right multiplication action of on itself. That is, form an edge in if there is such that . Another example is the graph of the left-shift action of on the space : recall that the left-shift action is defined by

for and . Observe that the Schreier graph of this actions is a Borel graph, when we endow the space with the product topology.

Our examples will come from a generalization of this graph. First note that if we replace by any other standard Borel space , the space still admits a Borel product structure with respect to which the Schreier graph of the left-shift action defined as above, is a Borel graph. The main idea is to start with a Borel graph and restrict the corresponding Schreier graph on to an appropriate subset on which the elements are homomorphisms from to . This allows us to control certain properties (such as chromatic number or hyperfiniteness) of the resulting graph by the properties if . More precisely:

Definition 3.1.

Let be a Borel graph and be a countable group with a generating set . Let be the restriction of to the set

We will refer to as the target graph, and we will denote the map by (note that the vertices of are labeled by the elements of ). It is clear from the definition that is a Borel graph with degrees at most and that is a Borel map. We can immediately make the following observation.

Proposition 3.2.
  1. The action of on has no fixed-points.

  2. admits a Borel homomorphism to . Thus,

Proof.

Let and . Note that as and are adjacent in it follows that and are adjacent in as is a homomorphism. Consequently, the map is a Borel homomorphism from to and as there are no loops in . ∎

The -regular tree .

In this paper we only consider the case of the group

together with the generating set . Since is isomorphic to the -regular infinite tree, , we use to denote the graph . Note also that we consider and equipped with a -edge coloring. As suggested above, an equivalent description of the vertex set of is that it is the set of pairs where is a homomorphism from the tree to and is a distinguished vertex of , a root. Then we have that and form an (-)edge if and only if and is an -edge in . This is because (a) there is a one-to-one correspondence between a homomorphism from to and the pairs , and (b) the shift action corresponds to changing the root for a fixed homomorphism from to .

Recall that an action is free if for each and we have . The free part, denoted by , is the set . Note that the left-shift action of on, say, is not free, in particular, the corresponding Schreier-graph has cycles. To remedy this, Marks used a restriction of the graph to the free part, showing that for each this graph has Borel chromatic number . Analogously, we have the following.

Definition 3.3.

Let that is, the restriction of the graph to the free part of the action.

In our first application we will use this remedy to get acyclic graphs. Note that for each edge in there is a unique generator that induces it. In particular, the graph admits a canonical Borel edge -coloring. The following is straightforward.

Proposition 3.4.

Let be a locally countable Borel graph. If is nonempty, then it is -regular and acylic.

However, utilizing the homomorphism graph together with an appropriate target graph, we will be able to completely avoid the non-free part, in an automatic manner. This way we will be able to guarantee the hyperfiniteness of the homomorphism graph as well. Recall that comes with a -edge coloring by the elements of . Let us consider the a subgraph of the homomorphism graph that arises by requiring to preserve this information.

Definition 3.5.

Assume that the graph is equipped with a Borel edge -labeling. Let be the restriction of to the set

Clearly, is also a Borel graph. Note that in the following statement the labeling of the edges of the target graph is typically not a coloring.

Proposition 3.6.

Assume that is an acyclic graph equipped with a Borel edge -labeling and is nonempty. Then

  1. is acyclic,

  2. If is hyperfinite, then so is ,

  3. is -regular.

Proof.

Observe that if is a homomorphism from a tree to an acyclic graph that is not injective, then there must be adjacent pairs of vertices and with and . Thus, if is edge label preserving then it must be injective, as incident edges have different labels in . Therefore, the map is injective on each connected component of , yielding (1).

To see (2), let be a witness to the hyperfiniteness of . Let be the pullback of by the map . Since is injective on every connected component, the graphs also have finite components and their union is .

For (3) just notice that using the injectivity of again, it follows that has cardinality . ∎

4 Variations on Marks’ technique

Now we are ready to adapt Marks’ technique [Mar16] to homomorphism graphs. Let us denote by the weakly provably -chromatic number of (see Section 2).

Theorem 1.1.

Let be a locally countable Borel graph. Then

The games we will define naturally yield elements rather than . In order to deal with the cyclic part of the graph, we will show slightly more, using the same strategy as Marks. Let , an anti-game labeling of is a map such that there are no and distinct vertices with and .

Remark 4.1.

One can define analogously anti-game labelings for graphs with with edges labeled by . Note that in the case when the graph is -regular and the labeling is an edge -coloring, the existence of an anti-game coloring is equivalent to solving the well known edge grabbing problem (that is, every vertex picks one adjacent edge but no edge can be picked from both sides); see also [BBE20].

Lemma 4.2.

There exists a Borel anti-game labeling .

Proof.

Let us use the notation . By definition, the action on every connected component is not free. Using [KM04, Lemma 7.3] we can find a Borel maximal family of pairwise disjoint finite sequences each of length at least such that for each there is a sequence such that , for , and , . (Note that it is possible that in which case there are two distinct generators such that .)

Now label an element by if for some . Otherwise, let be the minimal such that has strictly smaller distance to than with respect to the graph distance in . It is easy to check that is an anti-game labeling. ∎

Proof of Theorem 1.1.

We will show that there is no Borel anti-game labeling . Note that this yields Theorem 1.1, as if admitted a Borel -coloring, combining this with the coloring constructed in Lemma 4.2 we would obtain a Borel anti-game labeling on . So, towards contradiction, assume that is a Borel anti-game labeling.

Without loss of generality we may assume that has no isolated points. This ensures that the games below can be always continued.

We define a family of games parametrized by elements of , for two players. In a run of the game players and alternate and build a homomorphism from to , i.e., an element of , with the property that .

1st

2nd

Alice labels

Bob labels
Figure 1: The game

In the -th round, first labels vertices of distance from the on the side of the edge. After that, labels all remaining vertices of distance , etc (see Fig. 1). In other words, labels the elements of corresponding to reduced words of length starting with then labels the rest of the reduced words of length . For the reader familiar with Marks’ construction, let us point out that for a fixed , the games are analogues to the ones he defines, with the following differences: allowed moves are vertices of the target graph and restricted by its edge relation.

The winning condition is defined as follows:

Lemma 4.3.
  1. For any and one of the players has a winning strategy in the game .

  2. The set is weakly provably .

Proof.

Let us denote by the connected component equivalence relation of . Observe that as is connected, the range of any element is contained in a single class. By the Feldman-Moore theorem, there is a countable collection of Borel functions such that . Therefore, the games above can be identified by games played on , namely, labeling a vertex in by a vertex corresponds to playing the minimal natural number with . Since the functions are Borel, this correspondence is Borel as well. Moreover, the rule that must be homomorphism determines a pruned subtree of legal positions and the map is Borel. This yields that there exists a Borel set such that

Now, the first claim follows from the Borel determinacy theorem, while the second follows from Lemma 2.5. ∎

Claim 4.4.

For every there is an such that wins .

Proof.

Suppose not. Then we can combine strategies of for each in the natural way to build a homomorphism that is not in the domain of (see, e.g., [Mar] or [Mar16]). ∎

Now we can finish the proof of Theorem 1.1. Define by

(1)

Since weakly provably sets form an algebra, is weakly provably measurable and by Claim 4.4 it is everywhere defined. By our assumptions on there are adjacent with . Now, we can play the two winning strategies corresponding to games and of against each other, as if the first move of was (resp. ). This yields distinct homomorphisms with and , contradicting that is an anti-game labeling. ∎

4.1 Generalizations

Edge labeled graphs.

As mentioned above, a novel feature of our approach is that requiring the homomorphisms to be edge label preserving and ensuring that is acyclic, we can get rid of the investigation of the cyclic part (see Proposition 3.6). In order to achieve this, we have to assume slightly more about the chromatic properties of the target graph.

Assume that is equipped with and edge -labeling. The edge-labeled chromatic number, of is the minimal , for which there exists a map so that for each the set doesn’t span edges with every possible label. In other words, if and only if no matter how we assign many colors to the vertices of , there will be a color class containing edges with every label. We define measurable, Baire measurable, etc. versions of the edge-labeled chromatic number in the natural way.

Theorem 4.5.

Let be a locally countable Borel graph with a Borel -edge labeling, such that for every vertex and every label there is an -labeled edge incident to . Then

Proof.

The proof is similar to the proof of Theorem 1.1, but with taking the edge colors into consideration. Let us indicate the required modifications. We define as above, with the extra assumption that players must build a homomorphism that respects edge labels, i.e., an element . The condition on the edge-labeling ensures that the players can continue the game respecting the rules at every given finite step.

The analogue of Claim 4.4 clearly holds in this case, and we can define as in (1). Finally, guarantees the existence of and such that and that the edge between and has label , which in turn allows us to use the winning strategies of in and against each other, as above. ∎

Graph homomorphism.

In what follows, we will consider a slightly more general context, namely, instead of the question of the existence of Borel colorings, we will investigate the existence of Borel homomorphisms to a given finite graph . The following notion is going to be our key technical tool.

Definition 4.6 (Property -(*)).

Let and be a finite graph. We say that satisfies property -(*) if there are sets such that restricted to has chromatic number at most for , and there is no edge between vertices of and .

Note that implies -(*): indeed, if are independent sets that cover , we can set . Later, in Proposition 5.8 we show that there are graphs that satisfy -(*) and have chromatic number . This is best possible as, if a graph satisfies -(*) then . In order to see this, take witnessing -(*). Then, as there is no edge between and , so in particular, between and , it follows that the chromatic number of ’s restriction to is . But then we can construct proper -colorings of and , which shows our claim.

Theorem 4.7.

Let be a locally countable Borel graph and assume that is a finite graph with -(*). Assume that is equipped with a Borel -edge labeling, such that every vertex is incident to some edge with every label. Then

Proof.

Assume for contradiction that such a Borel homomorphism exists. We will need a further modification of Marks’ games. Let . For define the game as in the proof of Theorem 4.5, with the winning condition modified to

Observe that playing the strategies of against each other as in Claim 4.4 we can establish the following.

Claim 4.8.

For every and every sequence with there is some such that has no winning strategy in .

Now let be the powerset of the set . Of course, . Define a mapping by

As in Lemma 4.3, the map is weakly provably -measurable. By our assumption on , there is a subset on which is constant and spans an edge with each label.

Lemma 4.9.

Let and be sets such that there is no edge between points of and in . Then for every has no winning strategy in at least one of and . In particular, if is independent in then cannot have a winning strategy in .

Proof.

If there exists an for which and can be won by , then, as is constant on , this is the case for every . So we could find connected with an labeled edge so that has winning strategies in and . Then we can play the two winning strategies of against each other as in the proof of Theorem 1.1. This would yield elements in that form an -edge with , contradicting our assumption on and . ∎

To finish the proof of the theorem, fix the sets from the definition of the condition -(*), and take an arbitrary . By Lemma 4.9, we get that for one of them, say , has a winning strategy . Let be independent sets from the definition of -(*), i.e., with the property that . Using Lemma 4.9 again, we obtain that has a winning strategy in for each . This contradicts Claim 4.8.

5 Applications

In this section we apply the theorems proven before to establish our main results. We will choose a target graph using three prominent notions from descriptive set theory: measure, category and Ramsey property.

5.1 Complexity of coloring problem

First we will utilize the shift-graph on to establish the complexity results. Let us mention that it would be ideal to use the main result of [TV21] (i.e., that deciding the Borel chromatic number of graphs is complicated) directly and apply the map together with Theorem 1.1 to show that this already holds for acyclic bounded degree graphs. Unfortunately, since the mentioned theorem requires large weakly provably -chromatic number, this does not seem to be possible (the graphs constructed in [TV21] only have large Borel chromatic numbers, at least a priori). Instead, we will rely on the uniformization technique from [TV21]. Roughly speaking, the technique enables us to prove that in certain situations deciding the existence of, say, Borel colorings is -hard, whenever we are allowed to put graphs “next to each other”.

Let be uncountable Polish spaces, be a class of Borel sets and be a map. Define by and let the uniform family, , be defined as follows: for let

and

(that is, it contains the graph of a Borel function ).

Let be a family of subsets of Polish spaces. Recall that a subset of a Polish space is -hard, if for every Polish and there exists a continuous map with . A set is -complete if it is -hard and in . A family of subsets of a Polish space is said to be -hard on , if there exists a set so that the set is -hard. The next definition captures the central technical condition.

Definition 5.1.

The family is said to be nicely -hard on if for every there exist sets and so that and for all we have

A map is called on if for every Polish space and we have . Now we have the following theorem.

Theorem 5.2 ([Tv21], Theorem 1.6).

Let be uncountable Polish spaces, be a class of subsets of Polish spaces which is closed under continuous preimages, finite unions and intersections and . Suppose that is on and that is nicely -hard on . Then the family is -hard on .

Let us identify infinite subsets of with their increasing enumeration. If let us use the notation in the case the set is infinite and if it is co-finite. Set It follows form the fact that restricted to sets of the form