Reducts of the Random Bipartite Graph
Let be the random bipartite graph, a countable graph with two infinite sides, edges randomly distributed between the sides, but no edges within a side. In this paper, we investigate the reducts of that preserve sides. We classify the closed permutation subgroups containing the group , where is the group of all isomorphisms and anti-isomorphisms of preserving the two sides. Our results rely on a combinatorial theorem of Nešetřil-Rödl and a strong finite submodel property for .
As in [sT96], a reduct of a structure is a structure with the same underlying set as , for some relational language, each of whose relations is -definable in the original structure. If is -categorical, then a reduct of corresponds to a closed permutation subgroup in (the full symmetric group on the underlying set of ) that contains (the automorphism group of ). Two interdefinable reducts are considered to be equivalent. That is, two reducts of a structure are equivalent if they have the same -definable sets, or, equivalently, they have the same automorphism groups. There is a one-to-one correspondence between equivalence classes of reducts and closed subgroups of containing via (see [sT96]).
There are currently a few -categorical structures whose reducts have been explicitly classified. In 1977, Higman classified the reducts of the structure (see [gH77]). In 2008, Markus Junker and Martin Ziegler classified the reducts of expansions of by constants and unary predicates (see [mJ05]). Simon Thomas showed that there are finitely many reducts of the random graph ([sT91]) in 1991, and of the random hypergraphs ([sT96]) in 1996. In 1995 James Bennett proved similar results for the random tournament, and for the random -edge coloring graphs ([jB95]). In this paper, we investigate the reducts of the random bipartite graph that preserve sides. We find it convenient to consider a bipartite graph in a language with two unary predicates (one side , the other side ) and two binary predicates (edge , not edge ). Equivalently, we analyze the closed subgroups of containing , where , denote the two sides of the random bipartite graph. Let be a group of all isomorphisms and anti-isomorphisms preserving the two sides. We classified all the closed subgroup of containing . We have analyzed some closed groups between and but do not describe the results here since we do not have a classification of all such.
A structure = , where and , is a bipartite graph if it satisfies the following set of axioms:
In the rest of the paper, we will use the following notations: if , then we call a cross-edge, and we say that has cross-type if holds for the pair for . Furthermore, if and , we denote by . An subgraph is a bipartite graph with vertices in and vertices in . denotes the group .
Let . A bipartite graph satisfies the extension property if for any two disjoint subsets , , and any two disjoint subsets , ,
there exists a vertex such that for every for ; and
there exists a vertex such that for every for .
A countable bipartite graph, denoted by , is random if it satisfies the extension properties for every .
The ’s are first-order sentences, and the axioms in Definition 1.1 together with the form a complete and -categorical theory. A random bipartite graph can be built by Fraisse-construction for bipartite graphs (see [wH77]). It is countable and unique up to isomorphism. It is also easy to show that the random bipartite graph is homogeneous by a back-and-forth argument. In the rest of paper, we denote by the random bipartite graph.
Let be the random bipartite graph and be a subset of . A bijection is a switch with respect to if the following conditions are satisfied:
for all and , if and only if .
Note that a switch on any finite set of vertices can be obtained by composing single-vertex switches.
Let . The switch group is the closed subgroup of generated as a topological group by
The set of all such that is a switch with respect to some , where .
Since satisfies the extension property for and is closed, we can construct which is a switch w.r.t. . Observe that . Let be the closed group generated by and . Then the group is the same as the group except when . Notice , which is a group of permutations that either preserve all cross-types on , or exchange all cross-types on . Also notice that .
We now state the main result of this paper.
If is a closed subgroup with , then there exists a subset such that .
That is, there are only finitely many closed subgroups of containing : , and . This theorem relies on a combinatorial theorem of Nešetřil-Rödl and the strong finite submodel property of the random bipartite graph. It is still an open question whether there are finitely many closed subgroups between and .
Here is how the rest of the paper is organized. In section 2, we study the relations preserved by the groups , where . In section 3, we show that the random bipartite graph has the strong finite bipartite submodel property. In section 4, we employ a technique called -analysis for the random bipartite graph. These prepare us to give an explicit classification of the closed subgroups of containing in the rest of the paper. In section 5, we prove the first part of Theorem 1.6, which says that the closed subgroups of containing are , and , and . In section 6, we proved the existence of some special finite subgraphs of , which will be used in section 7. Then in section 7 we show there is no other proper closed subgroup between and , which completes the proof of Theorem 1.6.
2 Relations Preserved by Switch Groups
In this section, we identify the relations preserved by the switch groups and . For convenience in discussing closures of , we let .
Let , and be a finite bipartite subgraph of . We say preserves the parity of cross-types on if the number of cross-types in is even if and only if the number of cross-types in is even.
preserves the parity of cross-types in every -subgraph of .
It is easy to show that any preserves the parity of cross-types in every -subgraph of . The other direction is proved as follows.
Suppose preserves the parity of cross-types in every -subgraph of . Let B be an arbitrary -subgraph of . Since preserves the parity of ’s for and , only an even number of the cross-types can be changed. That is, , , or of the cross-types can be changed. We shall prove that in each case, there exists such that .
Case 1: if none of the cross-types are changed, then there exists such that .
Case 2: if two of the cross-types are changed, then there exists which is either a switch w.r.t. one vertex or a switch w.r.t. two vertices of B such that .
Case 3: if four of the cross-types are changed, then there exits which is a switch w.r.t. of (i.e. ) such that .
We then choose a vertex and let . We may assume . Note if is a cross-edge in and does not preserve the cross-type on , then for some . Also notice that and both preserve the parity of cross-types in -subgraphs of , hence so does . Then it is easy to check that either for every , ; or for every , , where and . Therefore , and so . Continuing in this manner for the vertices in , we see that for any finite bipartite graph , there exists an element such that . Thus , since is closed. This complete the proof of Lemma 2.2.
Similarly, we can prove the following results.
preserves the parity of cross-types in every -subgraph of .
preserves the parity of cross-types in every -subgraph of .
3 The Strong Finite Bipartite Submodel Property
In this section, we define the Strong Finite Bipartite Submodel Property (SFBSP), inspired by the Strong Finite Submodel Property introduced by Thomas in [sT96], and we prove that the random bipartite graph has the SFBSP. This property provides a powerful tool in the later sections of this paper.
A countable infinite bipartite graph has the Strong Finite Bipartite Submodel Property (SFBSP) if is a union of an increasing chain of substructures such that
and for each . In particular,
if is even, then ;
for any sentence with , there exists such that for all .
The countable random bipartite graph has the SFBSP.
Theorem 3.2 is a consequence of the Borel–Cantelli Lemma, as below:
Definition 3.3 ([sT96])
If is a sequence of events in a probability space, then is the event that consists of realization of infinitely many of , denoted by .
Lemma 3.4 (Borel–Cantelli, [pB79])
Let be a sequence of events in a probability space. If , then .
[Proof of Theorem 3.2] Since the extension properties ’s axiomatize the random bipartite graph and implies for all , for every sentence true in , there exists some such that holds if and only if holds. Let be the probability space of all countable bipartite graphs , where and every cross-edge has cross-type with probability . For each with , let such that if is even, then , otherwise . Let be the event that the induced graph on does not satisfy the extension property . Then by simple computation,
where is the number of combinations of objects taken at a time. Let . Then . By the ratio test for infinite series, we have converges, and so does . Thus by Lemma 3.4, . So there exists a bipartite graph and an integer such that for all , the subgraph on satisfies the extension property , and so . Notice that the choice of ensures that is countable and satisfies all the axioms for the random bipartite graph. Hence is isomorphic to . Then has the SFBSP, which completes the proof of Theorem 3.2.
In the rest of the paper, we often use the fact that has the usual finite submodel property. We will only use the strong finite bipartite submodel property in section .
In [sT96], Thomas used a helpful tool called ”m-analysis” to classify the reducts of the random hypergraphs. Using a similar approach, we give the definition of -analysis in this section, and we prove that if and if is sufficiently large, then has an -analysis. This rather technical concept will be used in the proof of Theorem 1.6.
Let . Suppose and satisfies and . An -analysis of consists of a finite sequence of elements satisfying the following conditions:
For each , there exist a finite -subgraph in , and an element such that
is either an automorphism, or a switch with respect to some vertex where ;
is an isomorphic embedding.
We now prove the existence of an -analysis for a given .
Let and . For every , there exists an integer such that if for and , then there exists an -analysis of .
Let be such that is a very large subset of . By Ramsey’s Theorem, there exists a large subset of such that satisfies one of the following two conditions for every cross-edge in , where :
We will construct a sequence of ’s as following.
If holds, then we let where is the identity map on . Let be an arbitrary -subgraph in , and choose such that . Define .
Next we choose if it exists, and consider . Since and is the identity map, is either an isomorphism or a switch with respect to by Lemma 2.2. Let be an arbitrary -subgraph of containing . Then there exists which is either an isomorphism or a switch with respect to and . Define .
Continuing in this manner, for , we can find an -subgraph of and such that
is either an isomorphism or a switch with respect to some vertex where ;
Also is an isomorphic embedding.
If holds, then there exists with , which exchanges all the cross-types on . Let . Hence is an isomorphism. The rest of the proof will be the same as in .
Hence is an -analysis of . This completes the proof of Theorem 4.2.
5 Closed Subgroups of Containing
In this section, we prove the first part of Theorem 1.6, which says that the closed subgroups of containing are , , , and . Notice that in the rest of the paper, we only consider maps in . Hence from now on, we call is an isomorphism if is a cross-edge and implies for . We call is an anti-isomorphism if is a cross-edge and implies for .
Suppose that is a closed subgroup with . Let be the largest subset of such that . Then , and so .
In the rest of this section, we let be a closed subgroup with , and be the largest subset of such that .
Suppose that is a bijection such that for every finite with for and , we have . Then .
Now let . Let be an arbitrary finite bipartite graph with for and . Then it will be sufficient to show that . To achieve this, we adjust repeatedly via composition with elements of until we eventually obtain an element such that is an isomorphism. Our strategy is based upon the following lemma.
Suppose , and , are two disjoint bipartite subgraphs such that for every cross-edge in , is an isomorphism. Then is an isomorphism.
We prove this by contradiction. Suppose is not an isomorphism, then there exists a cross-edge such that is not an isomorphism. Let be a -subgraph of such that . By assumption, is an isomorphism for every cross-edge . Thus does not preserve the parity of the cross-types on the -subgraph , which contradicts Lemma 2.2. This completes the proof of Lemma 5.3.
We shall make use of the following property of .
Let X be the largest subset of such that . There exists a finite bipartite subgraph of satisfying:
For any , if there exist some vertex and such that is a switch w.r.t , then .
We prove the equivalent statement: there exists a finite bipartite subgraph of satisfying: if and , then for every and every , is not a switch w.r.t .
Since and , there exists a map which is a switch with respect to some vertex , but not in . Otherwise the closed group generated by and is , and so is a subgroup of , a contradiction with the definition of . Then implies that for every , is not a switch with respect to . So there exists a finite set containing such that for every , is not a switch with respect to
Since has the extension property, we have the following holds:
For every vertex , there exists a bipartite graph containing which is isomorphic to mapping to . This can be expressed by the first-order sentence . If is the sentence , then . Hence by Theorem 3.2, there exists a finite bipartite of such that . This satisfies our requirement, which completes the proof of Lemma 5.4.
We shall also make use of a combinatorial theorem of Nešetřil–Rödl, which is a generalization of Ramsey’s Theorem. The following formulation, convenient for our use, is due to Abramson and Harrington ([fA78]).
Definition 5.5 (See [sT96])
A system of colors of length , is an -sequence of finite nonempty sets. An -colored set consists of a finite ordered set and a function such that for each where . For each , is called the color of . An -pattern is an -colored set whose underlying ordered set is an integer.
Theorem 5.6 (Abramson–Harrington [fA78])
Given , , , a system of colors of length and an -pattern , there exists an -pattern with the following property. For any -colored set with -pattern and for any function , there exists such that has -pattern and such that for any , depends only on the -pattern of . (We say that such is -homogeneous).
Fix an ordering of vertices in such that is an initial segment of this ordering of . For a suitable system of colors , we define an -coloring of by setting:
if and only if and the order-preserving bijection is an isomorphism.
Now we define the partition function such that for ,
if for ; or if with is an isomorphism.
Let be the finite bipartite graph given by Lemma 5.4 and let , . Since satisfies the extension properties, the following conditions hold.
for and , where as in Lemma 4.2;
contains all different copies of -graphs, each connecting to in all possible ways;
contains isomorphic copies of -subgraph connecting to in all possible ways.
For every , there exists a finite bipartite subgraph containing such that is isomorphic to the -subgraph .
These can be expressed as a first-order sentence . Since has SFBSP, there exists a finite subgraph such that the conditions hold in . Now let the -pattern P be the one derived from . By Theorem 5.6 there exists such that has the -pattern . Thus is isomorphic to sending to . Furthermore, is -homogeneous. Now we will use the following Claims.
Suppose that , and that for and . Let be an order-preserving bijection such that is an isomorphism for all . Then for all ,
is an isomorphism if and only if is an isomorphism.
We prove this by contradiction. We may assume that there exists some such that is an isomorphism while is not. Since satisfies condition , there exist -subgraphs , , and , with and satisfying the following condition.
There exists an order-preserving bijection mapping to such that for every , is an isomorphism.
In particular, for all . Since is -homogeneous, it follows that for all , is an isomorphism if and only if is an isomorphism. Since and , we have is an isomorphism but is not an isomorphism. Let is not an isomorphism and is not an isomorphism. Then because of the effect of on and . But by Lemma 2.2, implies preserve the parity of cross-types in and . Thus and must be even, which contradicts . This complete the proof of Claim A.
Since satisfies the condition , by Theorem 4.2 there exists an -analysis of : . That is, for each , there exists a finite -subgraph in and an element such that
is either an isomorphism or a switch w.r.t some vertex where ;
is an isomorphic embedding.
If all , then , and so . Otherwise, let be the least integer such that and the corresponding is a switch w.r.t. . Note , which implies . We prove this situation can not occur. Note that is a switch w.r.t a vertex .
Since satisfies the condition , there exist an -subgraph which is an isomorphic copy of H, and a map satisfying that is an order-preserving bijection such that is an isomorphism for all .
By Claim A, for every , is an isomorphism if and only if is an isomorphism. Next we will show there exist such that is a switch w.r.t of in . But then Lemma 5.4 implies that , contrary to our assumption. We define inductively for such that for all is an isomorphism if and only if is an isomorphism.
Suppose have been defined, we now define for :
If is an isomorphism, or if is a switch w.r.t. but , then is an isomorphism on , which is in . We define