Local Dimension is Unbounded for Planar Posets

In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Ne\v{s}et\v{r}il and Pudl\'ak in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. In just the last year, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width, and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height.


Notation, Terminology and Background Discussion
In this paper, we investigate combinatorial problems for finite posets. As has become standard in the literature, we use the terms elements and points interchangeably in referring to the members of the ground set of a poset. We will write x y in P when x and y are incomparable in a poset P , and we let Inc(P ) denote the set of all ordered pairs (x, y) with x y in P . As a binary relation, Inc(P ) is symmetric. Recall that a non-empty family R of linear extensions of P is called a realizer of P when x < y in P if and only if x < y in L for each L ∈ R. Clearly, a non-empty family R of linear extensions of P is a realizer of P if and only if for each (x, y) ∈ Inc(P ), there is some L ∈ R for which x > y in L. The dimension of a poset P , as defined by Dushnik and Miller in their seminal paper [3], is the least positive integer d for which P has a realizer R with |R| = d.
In recent years, researchers have been investigating combinatorial problems for two variations of the Dushnik-Miller concept for dimension, known as Boolean dimension and local dimension, respectively.
Here is the setting for Boolean dimension. For a positive integer d, we let 2 d denote the set of all 0-1 strings of length d. Such strings are also called bit strings. Let P be a poset and let B = {L 1 , L 2 , . . . , L d } be a family of linear orders on the ground set of P (these linear orders need not be linear extensions of P ). Also, let τ be a function which maps all 0-1 strings of length d to {0, 1}. For each pair (x, y) of distinct elements of P , we form the bit string q(x, y, B) which has value 1 in coordinate i if and only if x < y in L i . We call the pair (B, τ ) a Boolean realizer of P if for every pair (x, y) of distinct elements of P , x < y in P if and only if τ (q(x, y, B)) = 1. Nešetřil and Pudlák [12] defined the Boolean dimension of P , denoted bdim(P ), as the least positive integer d for which P has a Boolean realizer (B, τ ) with |B| = d. Clearly, bdim(P ) dim(P ), since if R = {L 1 , L 2 , . . . , L d } is a realizer of P , we simply take τ as the function which maps (1, 1, . . . , 1) to 1 while all other bit strings of length d are mapped to 0.
Trivially, bdim(P ) = 1 if and only if P is either a chain or an antichain. Furthermore, if Q is a subposet of P , then bdim(Q) bdim(P ). In this paper, we will denote the dual of poset P as P * . Clearly, bdim(P ) = bdim(P * ). It is an easy exercise to show that if bdim(P ) = 2, then dim(P ) = 2.
In [15], Trotter and Walczak prove the modestly more challenging fact that if bdim(P ) = 3, then dim(P ) = 3. As we will see shortly, for every d 4, there is a poset P with bdim(P ) = 4 and dim(P ) = d.
Here is the setting for local dimension. Let P be a poset. A partial linear extension, abbreviated ple, of P is a linear extension of a subposet of P . Whenever L is a family of ple's of P and u ∈ P , we set µ(u, L) = |{L ∈ L : u ∈ L}|. In turn, we set µ(P, L) = max{µ(u, L) : u ∈ P }. A family L of ple's of a poset P is called a local realizer of P if the following two conditions are satisfied: (1) If x < y in P , there is some L ∈ L for which x < y in L; (2) if (x, y) ∈ Inc(P ), there is some L ∈ L for which x > y in L. The local dimension of P , denoted ldim(P ), is then defined 1 to be the least positive integer d for which P has a local realizer L with µ(P, L) = d. Clearly, Ueckerdt's new concept resonated with participants at the workshop, and it served to stimulate renewed interest in Boolean dimension as well.
Trivially, ldim(P ) dim(P ) for all posets P . Also, ldim(P ) = 1 if and only if P is a chain; ldim(Q) ldim(P ) if Q is a subposet of P ; and if P * is the dual of P , then ldim(P * ) = ldim(P ). It is an easy exercise to show that if ldim(P ) = 2, then dim(P ) = 2. However, as we will soon see, for every d 3, there is a poset P with ldim(P ) = 3 and dim(P ) = d. 1 The concept of local dimension is due to Torsten Ueckerdt who shared his ideas with participants of the workshop on Order and Geometry held in Gułtowy, Poland, September 14-17, 2016. The principal result of this paper involves a construction for a family of posets for which local dimension is unbounded. The implications of our construction fall into four distinct categories: 1.1. Planar Posets. A poset P is planar if its order diagram can be drawn in the plane without edge crossings. If a poset is planar, then its cover graph is planar, although the converse does not hold in general.
For an integer n 2, the standard example S n is a height 2 poset with minimal elements A = {a 1 , a 2 , . . . , a n } and maximal elements B = {b 1 , b 2 , . . . , b n }. Furthermore, a i < b j in S n if and only if i = j (see Figure 1). As is well known, dim(S n ) = n, for all n 2. Furthermore, it is Figure 1. The Standard Example an easy exercise to show that S n is planar when 2 n 4 and non-planar when n 5.
In Figure 2, we show a construction due to Kelly [11] showing that for all n 5, the non-planar poset S n is a subposet of a planar poset K n . This specific figure is a diagram for K 6 , but it should be clear how we intend that the diagram should be presented for other values of n.
Then set B = {L 1 , L 2 , L 3 , L 4 }, and let τ : 2 4 → {0, 1} be defined by setting τ (1, 1, 0, 1) = (1, 1, 1, 0) = 1. The map τ sends all other strings to 0. Clearly, (B, τ ) is a Boolean realizer for K n . In [12], Nešetřil and Pudlák remarked that the posets in the Kelly construction have Boolean dimension at most 4, and they asked whether if the Boolean dimension of planar posets is bounded. It is clear from their presentation that they believed the answer should be "yes." However, this challenging question has remained open for nearly 30 years.
For local dimension, Ueckerdt [17] noted that ldim(S n ) 3 for all n 2. In fact, ldim(K 2 ) = 2 and ldim(K n ) = 3 for all n 3. Here's why. Suppose n 3. Let L 1 and L 2 be the linear extensions of K n defined just above. Then for each i = 1, 2, . . . , n, let M i be the ple whose ground set is In view of these observations, it is also natural to ask whether local dimension is bounded for the class of planar posets. Our construction will show that the answer is "no." 1.2. Components and Blocks. We assume that the reader is familiar with basic concepts of graph theory, including the following terms: connected and disconnected graphs; components; cut vertices; and k-connected graphs for an integer k 2. Recall that when G is a graph, a connected induced subgraph H of G is called a block of G when H is 2-connected and there is no subgraph H of G which contains H as a proper subgraph and is also 2-connected.
Here are the analogous concepts for posets. A poset P is said to be connected if its cover graph is connected. A subposet B of P is said to be convex if y ∈ B whenever x, z ∈ B and x < y < z in P . Note that when B is a convex subposet of P , the cover graph of B is an induced subgraph of the cover graph of P . A convex subposet B of P is called a component of P when the cover graph of B is a component of the cover graph of P . A convex subposet B of P is called a block of P , when the cover graph of B is a block in the cover graph of P .
As is well known, when P is a disconnected poset with components C 1 , C 2 , . . . , C t , for some t 2, dim(P ) = max{2, max{dim(C i ) : 1 i t}. Readers may note that the preceding observation is just a special case of the formula for the dimension of a lexicographic sum (see page 23 in [13]).
For local dimension, it is an easy exercise to show that ldim(P ) 2 + max{ldim(C i ) : 1 i t}, but we do not know whether this inequality is best possible.
The corresponding result for Boolean dimension is more complicated and is due to Mészáros, Micek and Trotter [5].
The inequality in Theorem 1.1 cannot be improved dramatically, since it is shown in [5] that for large d, there is a disconnected poset P with bdim(P ) = Ω(2 d /d) and bdim(C) d for every component C of Q.
The situation with blocks is more complex, even for Dushnik-Miller dimension. In [16], Trotter, Walczak and Wang prove the following result for Dushnik-Miller dimension. Boolean dimension behaves somewhat like Dushnik-Miller dimension with respect to blocks, as the following inequality is also proved in [5]. Again, this inequality cannot be improved dramatically, as it is shown in [5] that for large d, there is a poset P with bdim(P ) = Ω(2 d /d) and bdim(B) d for every block B of P .
Our construction will show that local dimension behaves quite differently with respect to blocks. We will prove: Theorem 1.4. For every d 1, there is a poset P such that ldim(P ) = d while ldim(B) 3 for every block B of P .
1.3. Structural Graph Theory. The first major result linking dimension with structural graph theory is due to Joret, Micek, Milans, Trotter, Walczak and Wang [8], who showed that for each pair (t, h) of positive integers, there is a least positive integer d(t, h) so that if P is a poset of height h and the tree-width of the cover graph of P is t, then dim(P ) d(t, h). A poset of height 1 is an antichain and has dimension at most 2, so it is of interest to study d(t, h) only when h 2. Trotter and Moore [14] showed that d(1, h) = 3 for all h 2, and Joret, Micek, Trotter, Wang, and Wiechert [9] showed that d (2, h) 1276 for all h 2. As is well known, the posets in the Kelly construction [11] have cover graphs with path-width at most 3, so d(t, h) goes to infinity with h when t 3. The best bounds to date in the general case are due to Joret, Micek, Ossona and Wiechert [10]: However, Felsner, Mészáros and Micek [6] proved that the Boolean dimension of a poset is bounded in terms of the tree-width of its cover graph, independent of its height. Formally, here is their result. Theorem 1.5. For every t 1, there is a least positive integer d(t) so that if P is a poset whose cover graph has tree-width t, then bdim(P ) d(t).
Barrera-Cruz, Prag, Smith, Taylor and Trotter [2] proved that the local dimension of a poset is bounded in terms of the path-width of its cover graph, independent of its height. Formally, here is their result. Theorem 1.6. For every t 1, there is a least positive integer d (t) so that if P is a poset whose cover graph has path-width t, then ldim(P ) d (t).
However, it is also showed in [2] that the analogue of Theorem 1.5 for local dimension is false: Theorem 1.7. For every d 1, there exists a poset P with ldim(P ) > d such that the cover graph of P has tree-width at most 3.
Our construction provides a an alternative proof of Theorem 1.7.

Bounded Boolean Dimension and Unbounded Local Dimension.
Trotter and Walczak prove that if P is a poset and ldim(P ) 3, then bdim(P ) 8442. However, for each d 4, they then proved that there is a poset P with ldim(P ) = 4 and bdim(P ) = d. They also show that if bdim(P ) = 3, then ldim(P ) = dim(P ) = 3. In [2], it is shown that for each d 1, there is a poset P with bdim(P ) 4 and ldim(P ) > d. Our construction provides another instance of a family of posets where Boolean dimension is bounded and local dimension is not.

Our Construction
For the remainder of the paper, whenever we discuss a pair (n, d), it will be assumed that n and d are integers with n 3 and d 1. Also, we do not distinguish between an element x and a sequence (x) of length 1.
Fix an integer n 2. Then for each d 1, we define a planar poset K(n, d) via a recursive process. As suggested by the notation, the poset K(n, 1) = K n , i.e., K(n, 1) is just the Kelly construction illustrated in Figure 2.
Now suppose that we have defined the planar poset K(n, d) for some d 1. Suppose further that we have a drawing without crossings of the order diagram of K(n, d) and that in this drawing, (b n ) is the highest point. To form K(n, d + 1), we take the drawing of K(n, 1) illustrated in Figure 2 and make the following changes: For each i = 1, 2, . . . , n, we take a suitably small scaling of the drawing of K(n, d) and identify the point (b n ) in K(n, d) with the point a i in K(n, 1). We change the label of the points in the copy of K(n, d) by prepending the symbol a i at the start of the sequence.
Our construction for n = 4 and d = 2 With the obvious requirements regarding scaling in mind, it is obvious that posets in the family K = {K(n, d) : n 2, d 1} are planar. We note that for each pair (n, d), a block of the poset K(n, d) is isomorphic to a subposet of the Kelly construction K n . Accordingly, if B is a block in K(n, d), then ldim(B) 3.
We also note that posets in the family K have tree-width at most 3. Here's why: As is well known, the cover graph of the graph K(n, 1) has path-width at most 3, so it has tree-width at most 3. One of the basic properties of treewidth is that the tree-width of a connected graph is just the maximum treewidth of its blocks. Since the blocks of K(n, d) are isomorphic to subposets of the Kelly construction K n , they have local dimension at most 3. However, it should be noted that the path-width of cover graphs in the family K is not bounded.
Let A = {a 1 , a 2 , . . . , a n } and B = {b 1 , b 2 , . . . , b n }. Elements of K(n, d) will be called core points if they are sequences of elements of A ∪ B, and we will let Core(n, d) denote the subposet of K(n, d) determined by the core points. Note that Core(n, d) will always be a proper subposet of K(n, d). Also, note that when Core(n, 1) is just the standard example S n .
For the arguments to follow, it is important for us to understand the structure of the subposet Core(n, d). We state the following elementary proposition for emphasis. Proposition 2.1. Let u and v be distinct elements of Core(n, d).
Note that a 7 < b 3 in S 8 .
We also note the subposet Core(n, d) has height 2. Furthermore, the minimal elements of Core(n, d) are those sequences of length n with all coordinates in A, while the maximal elements of Core(n, d) are those sequences of length at most n with all coordinates except the last in A.

The Local Dimension of Posets in the Class K
In this section, we will show that the local dimension of posets in the class K is unbounded. This shows: (1) local dimension is not bounded for the class of planar posets; (2) the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks; and (3) the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height. Finally, we will have given another example of a family of posets where Boolean dimension is bounded and local dimension is not.
The proof of our main theorem will require a special case of a result which has become known as the "Product Ramsey Theorem," appearing in the classic text [7] as Theorem 5 on page 113. However, we will use slightly different notation in discussing this result.
Given a finite set X and an integer k with 0 k |X|, we denote the set of all k-element subsets of X by X k . When T 1 , T 2 , . . . , T t are k-element subsets of X 1 , X 2 , . . . , X t , respectively, we refer to the product g = T 1 × T 2 × · · · × T t as a k t -grid in X 1 × X 2 × · · · × X t .
Here is a formal statement of the version of the Product Ramsey Theorem we will use in our argument.
Theorem 3.1. For every 4-tuple (r, t, k, m) of positive integers with m k, there is a least positive integer n 0 = PRam(r, t, k, m) with n 0 k such that if |X i | n 0 for every i = 1, 2, . . . , t, then whenever we have a coloring φ which assigns to each k t -grid g in X 1 × X 2 × · · · × X t a color φ(g) from a set R of r colors, then there is a color α ∈ R so that for each j = 1, 2, . . . , t, there is an m-element subset H j ⊆ X j such that φ(g) = α for every k t -grid g in H 1 × H 2 × · · · × H t .
Actually, we will only use the case where k = 1, and now the theorem becomes a multi-dimensional version of the pigeon-hole principle. Readers who would be interested in how this theorem is applied to combinatorial problems on posets when k 2 are encouraged to consult [16], [4] and [2]. Now we are ready to state and prove our main theorem. Proof. The theorem holds trivially for d 2, with n 1 = n 2 = 2. So for the balance of the argument, we fix a value of d with d 3. Since Core(n, d) is a subposet of K(n, d), it suffices to show that ldim(Core(n, d)) d, provided n is sufficiently large. The argument will proceed by contradiction, i.e., we will assume that L is a local realizer for Core(n, d) with µ(z, L) d − 1 for every z ∈ Core(n, d). We will then show that this leads to a contradiction, provided n is sufficiently large.
We will now describe a recursive procedure which consists of d steps. The procedure will utilize a rapidly growing sequence (p 1 , p 2 , . . . , p d ) of integers with p 1 = d. For the moment, we defer the explanation as to how this sequence is determined, but in time it will be clear that this is done by repeated applications of Theorem 3.1.
The key idea of the proof is presented in the following technical claim. Proof. We prove the claim by induction on m ∈ [d]. For m = 1, we must be concerned only about item (2) in this list by setting H i = [n] for each i = 1, . . . , d.
Here's how the inductive step is carried out. We will prove the statement of the claim for m + 1 and an arbitrarily large p d−m ∈ N. Let assume that it is true for m and for some fixed p d+1−m ∈ N which depends from p d−m . (How big p d+1−m must be will be described later.) In consequence we obtain an integer n and appropriate h 1 , .
. Also, for each j = 1, 2, . . . , m, we let L j denote the subfamily of L consisting of those ple's which reverse at least one incomparable min-max pair in U j × V j . Note that each L j is non-empty whenever p d+1−m /m 1. We observe that if 1 j < j m, then L j ∩ L j = ∅. This follows from the fact, that if (u, v) and (u , v ) are incomparable min-max pairs from U j × V j and U j × V j , respectively, then u < v and u < v in Core(n, d).
Since |M| = m − 1, it follows that there is some integer j 0 ∈ [m] so that M ∩ L j 0 = ∅. Let h m be any integer in H m,j 0 and W = {a h 1 } × · · · × {a h m−1 } × {a hm } × A(H m+1,j 0 ) × · · · × A(H d,j 0 ), It is worth noting that W ⊆ U j 0 ⊆ Min(Core(n, d)) and v ∈ V j 0 ⊆ Max(Core(n, d)). Moreover we observe that v w in Core(n, d) for all w ∈ W. It follows that for each w ∈ W , there is some ple L in L j 0 ⊆ L − M with w > v in L. Since there are at most d − 1 ple's in L in which v appears, this results in a coloring of the elements of W using at most d − 1 colors. Since there is a natural one-to-one correspondence between elements of W and 1 d−m grids in H m+1,j 0 × · · · × H d,j 0 , we are then in a position to apply Theorem 3.1.
In particular, given a value of p d−m , we will assume that p d−m+1 is large enough to guarantee that there is a ple L ∈ L j 0 ⊆ L − M and a family K m+1 , . . . , K d of sets with K i ⊆ H i,j 0 and |K i | = p d−m+1 for each i = m + 1, m + 2, . . . , d, such that w > v in L for all w ∈ {a h 1 } × · · · × {a hm } × A(K m+1 ) × · · · × A(K d ) ⊆ W.
We then add the ple L to M and set H i = K i for all i = m + 1, m + 2, . . . , d.
The final contradiction occurs as we invoke Claim 1 for m = d and p 1 = d. Now we have a subfamily M of L with |M| = d − 1. It follows that there are exactly d incomparable min-max pairs of the form (u, v) with Furthermore, these incomparable min-max pairs form a subposet of Core(n, d) which is isomorphic to the standard example S d . Therefore, there are (at least) d distinct ple's in L which reverse these pairs and at least one of these does not belong to M. This shows there is some u with µ(u, L) d. The contradiction completes the proof.