Tree-width and dimension
Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension of a finite poset is bounded in terms of its height and the tree-width of its cover graph.
In this paper, we investigate combinatorial problems involving finite graphs and partially ordered sets (posets), linking the well-studied concept of tree-width for graphs with the concept of dimension for posets. The following is our main result.
For every pair of positive integers, there exists a least positive integer so that if is a poset of height at most and the tree-width of the cover graph of is at most , then the dimension of is at most . In particular, we have .
The remainder of this paper is organized as follows. In the next section, we provide a brief summary of essential notation and terminology for posets and dimension. This is followed by an even more compact section on graphs and tree-width. These sections are included since we anticipate that many readers will be quite familiar with one of these topics but less so with the other. With these basics in hand, we discuss in Section 4 the background behind this line of research and the motivation for our principal theorem. The proof of our main theorem is given in Section 5, and we discuss some open problems in Section 6.
2. Posets and Dimension
A partially ordered set (here we use the short term poset) is a set equipped with a reflexive, antisymmetric and transitive binary relation . Elements of are called points and here we will also call them vertices, since we will often consider graphs whose vertex set is the set of elements of . When the poset is fixed throughout the discussion, we abbreviate the statement in by just writing . The notation means of course and . These notations are reversible in the obvious manner, i.e., means the same as .
We say covers (also is covered by ) when , and there is no point with . Also, we associate with a poset a cover graph having the same vertex set as . The cover graph of has an edge when one of and covers the other. A drawing (typically, we consider only drawings with straight line segments for the edges) of the cover graph of a poset is called an order diagram (also, a Hasse diagram) if the point in the plane corresponding to the point is higher than the point corresponding to the point when covers in . We show in Figure 1 order diagrams for two different posets, both with the same cover graph.
When and are distinct points in a poset , and either or , we say and are comparable. When and are distinct points in , and they are not comparable, we say they are incomparable and write . We use the notation for the set of all ordered pairs with .
An element in a poset is minimal, respectively maximal when there is no point with , respectively . When is a subset of a poset , the restriction of the binary relation to is a poset and we call this a subposet of . A poset is called a linear order (also a total order) if . When is a subposet of and is a linear order, it is customary to call a chain. The largest positive integer for which has a subposet on points which is a chain in is called the height of .
A poset is called an antichain if it has height , i.e., for all and with . The largest integer for which contains a subposet on points which is an antichain is called the width of . The classic theorem of Dilworth  asserts that a poset of width can be partitioned into chains. Dually, Mirsky  proved that a poset of height can be partitioned into antichains.
Let and be posets. We call a linear extension of when has the same ground set as , is a linear order, and in whenever in . A family of linear extensions of is called a realizer of if in if and only if in for each . Clearly, a family of linear extensions of is a realizer if and only if for each , there is some with such that in .
Dushnik and Miller  defined the dimension of , denoted , as the least positive integer for which has a realizer with . Evidently, if and only if is a linear order. Also, when is a non-trivial antichain, as evidenced by the realizer where is an arbitrary linear order on the ground set of and is the dual of , i.e., in if and only if in .
In , Hiraguchi used Dilworth’s theorem to show that the dimension of a poset never exceeds its width. Hiraguchi also proved that if is a poset on points with , then . Both these inequalities are tight, as witnessed by a family of posets called standard examples and first studied in . As these posets play an important role later in this paper, we include here some details on their structure and properties.
For , the standard example is a height poset with minimal elements and maximal elements . The relation is defined on by setting if and only if , for all . For each , the width of is so . On the other hand, . This follows from the observation that if is a linear extension of , there can only be one integer with and in . Moreover, when , it is easy to see that is -irreducible, i.e., removing any point from lowers the dimension to .
There is a natural notion of isomorphism for posets, and it obvious that isomorphic posets have the same dimension. So it is natural to say that a poset contains a poset when there is a subposet of which is isomorphic to . In this vein, a poset has large dimension when it contains a large standard example. But this is far from necessary.
A poset is called an interval order when there is a family of closed intervals of the real line so that in if and only if in . Fishburn  showed that a poset is an interval order if and only if it does not contain the standard example . In , Füredi, Hajnal, Rödl and Trotter show that the maximum dimension of an interval order of height is . In particular, note that in order for an interval order to have large dimension, it must have very large height.
The standard examples show that in general, large height is not necessary for large dimension, and in , Felsner, Li and Trotter show that for every pair of positive integers, there is a height poset with so that the girth of the cover graph of is at least . The posets resulting from this construction contain but they do not contain , when .
Although cover graphs are useful in providing diagrams of posets, they do not seem to tell us much about the combinatorial properties of the posets associated with them. For example, the two posets shown in Figure 1 have the same cover graph. However, the poset on the left has height , width and linear extensions, while the poset on the right has height , width and linear extensions. Both posets have dimension .
At the extreme, a linear order on points has height , width and of course, a unique linear extension. However, when , the associated cover graph is bipartite, and the height poset with this same cover graph is called a fence. Now the width is and the number of linear extensions is exponentially large in . On the other hand, the dimension of a fence is when , so based only on these observations, one might conjecture that posets with the same cover graph have approximately the same dimension. But even this is not true. Later in the paper, we will show that for each , there are two posets having the same cover graph, one having dimension and the other having dimension at least .
However, there is another natural way to associate a graph with a poset. Like the cover graph, the comparability graph of has the same vertex set as but now we make an edge if and are comparable. The comparability graph of a poset contains the cover graph as a subgraph. Furthermore, if and are posets with isomorphic comparability graphs, then they have the same height, width, number of linear extensions and dimension. The fact that they have the same height and width is immediate. The fact that they have the same number of linear extensions and the same dimension follows in a straightforward manner from the pioneering work of Gallai  on comparability graphs.
With these remarks in mind, and with no additional background information to suggest otherwise, the principal result of this paper would then have to be viewed as a surprise.
3. Graphs and Tree-Width
In this paper, we consider only finite graphs without loops or multiple edges, and we assume that readers are familiar with basic concepts such as trees, paths, cycles, complete graphs, subgraphs, induced subgraphs, components, chromatic number, girth, genus, distance and diameter. Given a graph , an induced subgraph of is determined entirely by its vertex set. In particular, when is a tree, we will identify subtrees of just by specifying their vertex sets. So when and are subtrees of a tree , the statement just means that and have one or more vertices of in common.
Let be a graph with vertex set and edge set . The tree-width111We refer the reader to the text by Diestel  for a concise exposition of some of the key concepts behind this parameter. Diestel also provides interesting details on its history and the twenty year time period spanned by Robertson and Seymour’s proof of the Graph Minor Theorem. Also our notation for tree-width and some of our examples are taken from exercises in this text. of is the least positive integer for which there is a tree and a family of non-empty subtrees of so that
for all vertices in , ,
for all .
Trivially, a graph has tree-width if and only if it has no edges, while the tree-width of the complete graph on vertices is for all . Furthermore, if is a tree with at least one edge, then the tree-width of is . To see this, simply subdivide each edge in by inserting a new vertex in the interior of . Let denote the resulting tree. Then for each , take as the subtree of with vertex set (each is a star). Conversely, it is easy to see that a graph has tree-width at most if and only if it is acyclic.
Consider the following three basic operations on a graph: (1) delete an edge; (2) delete a vertex; (3) contract an edge. Given a graph , any graph that can be obtained from by applying a sequence of these basic operations is called a minor of . The following fundamentally important theorem, called the Graph Minor Theorem, is due to Robertson and Seymour 222The proof given by Robertson and Seymour for the Graph Minor Theorem appears in a series of papers published over the time span 1983 through 2004, and we cite here the culminating paper in that series..
If is an infinite sequence of graphs, then there are integers and with so that is isomorphic to a minor of .
A class of graphs is minor-closed if is in whenever is in and is isomorphic to a minor of . Examples of minor closed classes of graphs include the family of all planar graphs and, more generally, for fixed , the family of all graphs having genus at most . Also, it is easy to see that for each , the class of all graphs having tree-width at most is minor-closed.
Any proper minor-closed class of graphs admits a characterization by “forbidden minors”, i.e., a minimum family of graphs such that a graph belongs to if and only if it does not contain a minor isomorphic to a graph in . By the Graph Minor Theorem, the class is finite. The classic theorem of Wagner  asserts that the list of forbidden minors for the class of planar graphs consists of the complete graph and the complete bipartite graph .
Planar graphs can have large tree-width. Note that any bipartite graph is both the cover graph and the comparability graph of a height poset. In particular, the planar grid is bipartite and has tree-width (see Diestel , Exercises 14 and 21 on page 369). However, the tree-width of a planar graph is bounded in terms of its diameter333This result is implicit in the work of Baker  and made explicit by Bodlaender in .. Classes of graphs where tree-width is bounded in terms of diameter are said to satisfy the diameter tree-width property (also called the bounded local tree-width property).
The concept of path-width for graphs is defined just like tree-width except that it is required that the tree be a path, and of course the subtrees of are then just subpaths of . Trivially, the tree-width of a graph is at most its path-width. However, the tree-width of an outerplanar graph is at most (this follows from the observation that in a maximal outerplanar graph, there is always a vertex of degree two such that the neighbors of are adjacent to each other). On the other hand, outerplanar graphs can have arbitrarily large path-width. In fact, trees can have arbitrarily large path-width (see Diestel , Exercise 31 on page 370).
4. Background and Motivation
A poset is planar if its order diagram can be drawn without edge crossings in the plane. In Figure 2, we show on the left the order diagram of a height nonplanar poset. However, the cover graph of this poset is planar as witnessed by the drawing on the right.
We note that if is a height poset, then is planar if and only if its cover graph is planar [25, 3]. We also note that it is NP-complete to test whether a poset is planar , while there are linear-time algorithms for testing whether a graph is planar . Also, it is NP-complete to test whether a graph is a cover graph [6, 26].
When is a poset with only one minimal element, this single element is usually called a zero. Similarly, in a poset with only one maximal element, this element is called a one. The first result linking planarity and dimension is the following theorem of Baker, Fishburn and Roberts .
If is a planar poset with a zero and a one, then .
Subsequently, Trotter and Moore  proved the following extension.
If is a planar poset with a zero or a one, then .
Trotter and Moore  also obtained the following result as an immediate corollary to the preceding theorem.
If is a poset whose cover graph is a tree, then .
With the benefit of hindsight, one can argue that the line of research carried out in this paper might reasonably have been triggered 35 years ago, based solely on possible extensions to Corollary 4.3.
It is an easy exercise to show that the standard example is planar when , and as a consequence, there are -dimensional planar posets. On the other hand, is non-planar when . For a brief time in the late 1970’s, it was thought that it might be the case that whenever is a planar poset.
We pause here to answer a question raised earlier concerning the dimension of posets with the same cover graph. Specifically, we show that for each , there are posets and with the same cover graph with and . First, we consider a poset formed by modifying Kelly’s example as follows. For each , we add two new minimal points and with covered by and , while is covered by and . Clearly, is a subposet of so that .
On the other hand, there are exponentially many posets having the same cover graph as . One of them, which we denote , has and , for each . Obviously, both and are planar poset as witnessed by trivial modifications to the diagram for given in Figure 3. Moreover, in , the point is now a zero and the point is now a one. So by Theorem 4.1, .
Returning to the general subject of the dimension of posets with planar cover graphs, Felsner, Li and Trotter  proved the following result in 2010:
Let be poset of height . If the cover graph of is planar, then .
Actually, this was obtained as an easy corollary to the following theorems of Brightwell and Trotter [8, 7], published in 1997 and 1993, respectively (a new and quite elegant proof of this result has just been obtained by Felsner ).
Let be a planar multi-graph and let be the vertex-edge-face poset determined by a drawing without edge crossings of in the plane. Then . Furthermore, if is a simple, -connected planar graph, then the subposet determined by the vertices and faces is -irreducible.
The inequality in Theorem 4.4 is best possible as evidenced by the standard example . Noting that the poset in Kelly’s construction has height , Felsner, Li and Trotter  conjectured the following generalization, which was proved by Streib and Trotter  in 2012.
For every positive integer , there is a least positive integer so that if is a poset with a planar cover graph and the height of is at most , then .
We have and . For , the upper bound on the constant produced in the proof of Theorem 4.6 is very large, as several iterations of Ramsey theory are used. From below, it is straightforward to modify Kelly’s original construction and decrease the height to . This can be accomplished by deleting , , and and relabelling , , and as , , and , respectively. Wiechert  constructed a planar poset of height with ; however, this construction does not seem to generalize for larger values of . Accordingly, when , we do not know whether there is a planar poset of height with . On the other hand, Streib and Trotter  showed that for each , there is a poset of height with so that the cover graph of is planar.
Theorems 4.1 and 4.2, as well as Corollary 4.3 all provide conditions where the dimension of a planar poset can be bounded independent of its height. In , Felsner, Trotter and Wiechert gave the following additional results of this nature.
Let be a poset.
If the cover graph of is outerplanar, then .
If the comparability graph of is planar, then .
Both inequalities in Theorem 4.7 are best possible. The proof of the first inequality in Theorem 4.7 is relatively straightforward, but it takes a bit of work to show that it is best possible. However, the second inequality in Theorem 4.7 is quite different, and now the argument depends on the full strength of the Brightwell-Trotter inequality for the dimension of the vertex-edge-face poset determined by a drawing of a planar multi-graph, with the edges now playing a key role.
To the best of our knowledge, the following observation concerning Kelly’s 1981 construction was not made until 2012: The cover graphs of the posets in this construction have bounded tree-width. In fact, they have bounded path-width. We leave the following elementary observations as an exercise.
Let , let be the poset illustrated in Kelly’s construction, and let be the cover graph of . Then the height of is , and the path-width of is at most . In fact, when , contains as a minor, so its path-width is exactly .
We made some effort to construct large dimension posets with bounded height and cover graphs having bounded tree-width and were unable to do so. So consider the following additional information:
A poset whose cover graph has tree-width has dimension at most .
A poset whose cover graph is outerplanar has dimension at most . As noted previously, outerplanar graphs can have arbitrarily large path-width, but they have tree-width at most .
On the one hand, the tree-width of the cover graph of a planar poset can be arbitrarily large, even when the height of is . As an example, just take a height poset whose cover graph is an grid. On the other hand, the proof given by Streib and Trotter  to show that the dimension of a poset with a planar cover graph can be bounded in terms of its height used a reduction to the case where the cover graph of the poset is both planar and has diameter bounded in terms of the height of the poset. Again, as noted previously, a planar graph of bounded diameter has bounded tree-width.
Taking into consideration this body of evidence together with our inability to prove otherwise, it is natural to conjecture that the dimension of a poset is bounded in terms of its height and the tree-width of its cover graph, and this is what we now prove.
5. Proof of the Main Theorem
A subset of is said to be reversible if there is a linear extension of with in for every . It is then immediate that is the least positive integer so that there is a partition with each reversible. In view of this formulation, it is handy to have a simple test to determine whether a given subset of is reversible.
Let . An indexed subset of is called an alternating cycle when in for each , where we interpret the subscripts cyclically (i.e., we require in ). Reversing an alternating cycle would require a linear extension in which the cyclic arrangement alternates between strict inequalities of the form (needed to reverse ) and inequalities of the form (forced by ). Consequently, alternating cycles are not reversible. The following elementary lemma, proved by Trotter and Moore in  using slightly different terminology, states that alternating cycles are the only obstruction to being reversible.
If is a poset and , then is reversible if and only if contains no alternating cycle.
For the remainder of this section, we fix integers and , assume that is a poset with height and cover graph , and assume that the tree-width of is . Of course, we may also assume that . The remainder of the argument is organized to show that we can partition the set into reversible sets, where is bounded in terms of and .
Let denote the ground set of , so that is also the vertex set of the cover graph . Since the tree-width of is , there is a tree and a family of subtrees of such that (1) for each vertex of , the number of elements of with is at most , and (2) for each edge of , we have .
Let be the intersection graph determined by the family of subtrees of (some researchers refer to as the chordal completion of ). Evidently, the tree-width of is , and every edge of is an edge of . Of course, the set is also the vertex set of . In the discussion to follow, we will go back and forth, without further comment, between referring to members of as elements of the poset and as vertices in the cover graph and the intersection graph .
To help distinguish between vertices of and elements of , we will use the letters , , and (possibly with subscripts) to denote vertices of the tree , while the letters , and (again with subscripts) will be used to denote members of . The letters , , , , and will denote non-negative integers with the meaning of fixed by setting . The Greek letters and will denote proper colorings of the graph . The colors assigned by will be positive integers, while the colors assigned by will be sets of triples. Later, we will define a function which assigns to each incomparable pair a signature, to be denoted . We will use the Greek letter to denote a signature. The number of signatures will be the value , and we will use Lemma 5.1 to show that any set of incomparable pairs having the same signature is reversible. Of course, we must be careful to insure that is bounded in terms of and .
We consider the tree as a rooted tree by taking an arbitrary vertex of as root. Draw the tree without edge crossings in the canonical manner. The root is at the bottom, and each vertex that is not the root has a unique neighbor below—its parent (equipped with such a drawing, is called a planted tree). We suggest such a drawing in Figure 4.
For each , let denote the root of the subtree , i.e., the unique vertex of that is closest to the root of . Expanding vertices of if necessary, we may assume that whenever and are distinct elements of .
The tree may be considered as a poset by setting in when lies on the path from to in . Let denote the depth-first, left-to-right search order of . Let denote the depth-first, right-to-left search order of . It follows that in if and only if in and in 444Note that the poset obtained by adding a one to is planar. Now the argument given in  implies that , as evidenced by these two linear extensions.. This shows with unless . It is natural to say that is left of in , when in and in . Also, we say that is below in when in . When and are distinct elements of , exactly one of the following four statements holds: (1) is below in ; (2) is below in ; (3) is left of in ; and (4) is left of in .
The lowest common ancestor of two vertices and of , denoted , is the greatest vertex with and in .
5.2. Induced Paths in the Intersection Graph
Observe that is an edge of the graph if and only if one of the following statements is true: (1) in and , (2) in and .
We write when there is a sequence of elements of such that , , , and for each . Note that a shortest such sequence is an induced path in the graph . Therefore, we could alternatively have written this definition as follows: when there is an induced path in with , , , and in . As it will turn out, our proof will use the relation for .
The relation has the following properties:
if and , then ,
if and only if ,
if and only if there exists with and ,
if , then in ,
if and in , then ,
if and , then or ,
for each , .
Properties (1)–(4) follow directly from the definition of . To see (5), let be a sequence of elements of such that , , , and for each . Now note that since is a path in and , the union is a subtree of containing the path from to . In particular, , so there must be a positive with , and witnesses . To see (6), observe that and imply or in , and the conclusion follows from (5). Finally, the fact that is the tree-width of yields for each , whence (7) follows. ∎
We will use the properties listed in Lemma 5.2 implicitly, without further reference.
If in , then there exists such that:
Since contains the cover graph of , there is a path in with , , and in . Take the shortest such path. Since is the height of , we have . For each with , since is an edge of , we have when in or when in . If there is an index with and in , then we have , so is an edge of and we can obtain a shorter path by removing . Therefore, there is a unique index with in . The definition of yields and . Since , the conclusion follows for . ∎
5.3. Colorings of the Ground Set
Order the elements of as so that the following holds: if in , then . In particular, we have whenever . Define a coloring of with positive integers using the following inductive procedure. Start by setting . Thereafter, for , let be the least positive integer that does not belong to and . The reason why we take in this definition will become clear at the very end of the proof. The number of colors used by is at most . Actually, we are not that interested in how many colors will use exactly, except that this number must be bounded in terms of and , which it is.
If , and , then .
Suppose . Since and , we have or . Whichever of these holds, the definition of yields . ∎
Let be a pair of elements of with . There are four cases of how and are related in : (1) , (2) , (3) , or (4) . We associate with a triple , where is the number in denoting which of the above four cases holds. Since the number of distinct colors used by is bounded in terms of and , so is the number of distinct triples of the form for all pairs considered.
We define a new coloring of by assigning to each element of , the family and . Thus the colors used by are sets of triples, and the number of distinct colors used by is bounded in terms of and . Note that the color classes of refine the color classes of , as the first element of each triple in is , and is non-empty since .
If , and , then:
in if and only if in ,
in if and only if in .
Since and , there is with and . In particular, we have , which implies in view of Lemma 5.4. The conclusion now follows from . ∎
5.4. Signatures for Incomparable Pairs
Each incomparable pair in satisfies exactly one of the following six conditions:
is below in ,
is below in ,
is left of in and is left of in for each with and in ,
is left of in and there exists with , in and not left of in ,
is left of in and is left of in for each with and in ,
is left of in and there exists with , in and not left of in .
We define the signature of to be the triple , where is the number in denoting which of the above six cases holds for . Since the number of distinct colors used by is bounded in terms of and , so is the number of distinct signatures.
Let . To finish the proof of our main theorem, we show that is reversible for each signature . We argue by contradiction. Fix a signature , and suppose that is not reversible. In view of Lemma 5.1, contains an alternating cycle . Since all the signatures are equal, we have all the equal and all the equal. Moreover, all the pairs satisfy the same one of the conditions (1)–(6) above. This gives us six cases to consider. Case (2) is dual to (1), (5) is dual to (3), and (6) is dual to (4). Therefore, it is enough that we show that each of the cases (1), (3) and (4) leads to a contradiction. In the arguments below, we always interpret the index cyclically in .
Suppose that (1) holds for all . There must be an index such that is not below in . We have in , so let be an element of claimed by Lemma 5.3 for . Thus in , , and . Since is below and not below in , we have in and thus . We also have . Consequently, by Lemma 5.5, we have in , which is a contradiction.
If (3) holds for all , then we have left of in for each , which is clearly a contradiction.
Finally, suppose that (4) holds for all . There must be an index such that is not left of in . To simplify the notation, let , and . Thus we have and in , , left of in , and not left of in . Furthermore, since satisfies condition (4), there is with , in and not left of in . All this implies that the paths in connecting to and to both pass through . Now, let be an element of claimed by Lemma 5.3 for , and be an element of claimed by Lemma 5.3 for . Thus we have and in , , , , and . Since and in , it follows that is below every vertex in the path from to , and in particular, . Similarly, in . Thus or in . If , then and thus . This, by Lemma 5.5, implies in , which is a contradiction. If , then we get a similar contradiction . This completes the proof of Theorem 1.1.
6. Questions and Problems
Our main result establishes the existence of the function without emphasis on optimizing our bound. Let be the number of colors used in . The number of signatures of incomparable pairs is at most . We compute , and it follows that . One immediate challenge is to tighten the bounds on this function. It may even be true that for each , there is a constant so that . It is conceivable that better techniques may prove an exact formula for , for all and .
As noted in the introductory section, when the tree-width of the cover graph of is , , independent of the height of . Also, when the cover graph of is outerplanar (so it has tree-width at most ), independent of the height of . On the other hand, the posets in Kelly’s construction have path-width . Accordingly, it is natural to raise the following questions.
Does there exist a constant so that if is a poset and the path-width of the cover graph of is at most , then ?
Does there exist a constant so that if is a poset and the tree-width of the cover graph of is at most , then ?
The first of these two questions was recently settled in the affirmative by Biró, Keller and Young , and we firmly believe that the second one has an affirmative answer as well.
Kelly’s construction actually raises two other questions. First, is it true that a planar poset with large dimension contains a large standard example? We believe the answer is yes and make the following conjecture.
For every integer , there is an integer so that if is a planar poset with , then contains the standard example .
Second (and this specific question was posed to us by Stanley ), is it true that a planar poset with large dimension has many minimal elements? The answer is yes. Recently, Trotter and Wang  proved the following result.
If is a planar poset with minimal elements, then .
This inequality is best possible for and , but for larger values of , a lower bound of is proved in .
The first of these two questions has a natural extension to tree-width, so we would also make the following conjecture.
For every pair of positive integers with , there is an integer so that if is a poset such that the tree-width of the cover graph of is at most and , then contains the standard example .
While the second question concerning the number of minimal elements makes sense, it is easily answered in the negative, since adding a zero to a poset can increase the tree-width of the cover graph by at most one.
Finally, we close with what we believe is a very ambitious conjecture.
Let be a proper minor-closed class of graphs. Then for every integer , there is a least positive integer so that if is a poset of height and the cover graph of belongs to , then .
Our main theorem shows that the conjecture is true when is the class of graphs of tree-width at most . In , a general reduction is described which allows one to restrict to the case where the cover graph has bounded diameter (as a function of the height). It follows as an immediate corollary that the conjecture holds whenever has the diameter tree-width property. For this reason, we have an alternative proof of Theorem 4.6. Graphs of bounded genus, and more generally graphs excluding an apex graph as a minor also have the diameter tree-width property (see ). Therefore, the above conjecture also holds in these special cases.
The proof of our main theorem has been developed by the authors at a series of meetings at conferences and campus visits, as well as through email. However, we have also received valuable input from several other colleagues in our respective university environments. We would also like to thank anonymous referees who made helpful suggestions regarding the exposition and organization of material in this paper.
The last five authors gratefully acknowledge that the first author, Gwenaël Joret, is solely responsible for conjecturing our main theorem.
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