Applications of the Probabilistic Method to Partially Ordered Sets

Summary

There are two central themes to research involving applications of probabilistic methods to partially ordered sets. The first of these can be described as the study of random partially ordered sets. Among the specific models which have been studied are: random labelled posets; random t-dimensional posets; and the transitive closure of random graphs. A second theme concentrates on the adaptation of random methods so as to be applicable to general partially ordered sets. In this paper, we concentrate on the second theme. Among the topics we discuss are fibers and co-fibers; the dimension of subposets of the subset lattice; the dimension of posets of bounded degree; and fractional dimension. This last topic leads to a discussion of Ramsey theoretic questions for probability spaces.

1991 Mathematics Subject Classification. 06A07, 05C35.

Key words and phrases. Partially ordered set, poset, graph, random methods, dimension, fractional dimension, chromatic number Research supported in part by the Office of Naval Research.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Probabilistic methods have been used extensively throughout combinatorial mathematics, so it no great surprise to see that researchers have applied these techniques with great success to finite partially ordered sets. One central theme to this research is to define appropriate definitions of a random poset, and G. Brightwell’s excellent survey article [1] provides a summary of work in this direction.

A second theme involves the application of random methods to more general classes of posets. After this brief introductory section, we present four examples of this theme. The first example is quite elementary and involves fibers and co-fibers, concepts which generalize the notions of chains and antichains. The principal result here is an application of random methods to provide a non-trivial upper bound on the minimum size of fibers.

Our second example is more substantial. It involves the dimension of subposets of the subset lattice, an instance in which many of the classic techniques and results pioneered by Paul Erdős play major roles. The third example involves an application of the Lovász Local Lemma and leads naturally to the the investigation of the dimension of a random poset of height two.

Our last example involves fractional dimension for posets—an area where there are many attractive open problems. This topic leads to natural questions involving Ramsey theory for probability spaces.

The remainder of this section is a very brief condensation of key ideas and notation necessary for the remaining five sections. In this article, we consider a partially ordered set (or poset) P = (X, P) as a discrete structure consisting of a set X and a reflexive, antisymmetric and transitive binary relation P on X. We call X the ground set of the poset P, and we refer to P as a partial order on X. The notations x ≤ y in P, y ≥ x in P and (x, y) ∈ P are used interchangeably, and the reference to the partial order P is often dropped when its definition is fixed throughout the discussion. We write x < y in P and y > x in P when x ≤ y in P and x≠y. When x, y ∈ X, (x, y)∉P and (y, x)∉P, we say x and y are incomparable and write x ∥ y in P.

Although we are concerned almost exclusively with finite posets, i.e., those posets with finite ground sets, we find it convenient to use the familiar notation $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$ and $\mathbb{N}$ to denote respectively the reals, rationals, integers and positive integers equipped with the usual orders. Note that these four infinite posets are total orders; in each case, any two distinct points are comparable. Total orders are also called linear orders, or chains. We use n to denote an n-element chain with the points labelled as 0 < 1 < ⋯ < n − 1.

A subset A ⊆ X is called an antichain if no two distinct points in A are comparable. We also use P + Q to denote the disjoint sum of P and Q.

In the remainder of this article, we will assume that the reader is familiar with the basic concepts for partially ordered sets, including maximal and minimal elements, chains and antichains, sums and cartesian products, comparability graphs and Hasse diagrams. For additional background information on posets, the reader is referred to the author’s monograph [23], the survey article [14] on dimension by Kelly and Trotter and the author’s survey articles [21, 22, 25] and [26]. Another good source of background information on posets is Brightwell’s general survey article [2].

2 Fibers and Co-fibers

The classic theorem of Dilworth [4] asserts that a poset P = (X, P) of width n can be partitioned into n chains. Also, a poset of height h can be partitioned into h antichains. For graph theorists, these results can be translated into the simple statement that comparability graphs are perfect. Against this backdrop, researchers have devoted considerable energy to generalizations of the concepts of chains and antichains. Here is one such example.

Let P = (X, P) be a poset. Lone and Rival [18] called a subset A ⊆ X a co-fiber if it intersects every non-trivial maximal chain in P. Let cof(P) denote the least m so that P has a co-fiber of cardinality m. Then let cof(n) denote the maximum value of cof(P) taken over all n-element posets. In any poset, the set A ₁ consisting of all maximal elements which are not minimal elements and the set A ₂ of all minimal elements which are not maximal are both co-fibers. As $A_{1} \cap A_{2} = \varnothing$, it follows that cof(n) ≤ ⌊n ∕ 2⌋. On the other hand, the fact that cof(n) ≥ ⌊n ∕ 2⌋ is evidenced by a height 2 poset with ⌊n ∕ 2⌋ minimal elements each of which is less than all ⌈n ∕ 2⌉ maximal elements. So $\mathrm{cof}(n) = \lfloor n/2\rfloor$ (this argument appears in [18]).

Dually, a subset B ⊆ X is called a fiber if it intersects every non-trivial maximal antichain. Let fib(P) denote the least m so that P has a fiber of cardinality m. Then let fib(n) denote the maximum value of fib(P) taken over all n-element posets. Trivially, fib(n) ≥ ⌊n ∕ 2⌋, and Lone and Rival asked whether equality holds.

In [6], Duffus, Sands, Sauer and Woodrow showed that if P = (X, P) is an n-element poset, then there exists a set F ⊆ X which intersects every 2-element maximal antichain so that | F | ≤ ⌊n ∕ 2⌋. However, B. Sands then constructed a 17-point poset in which the smallest fiber contains 9 points. This construction was generalized by R. Maltby [19] who proved that for every ε > 0, there exist a n ₀ so that for all n > n ₀ there exists an n-element poset in which the smallest fiber has at least $(8/15-\epsilon )n$ points.

From above, there is no elementary way to see that there exists a constant α > 0 so that fib(n) < (1 − α)n. However, this is an instance where random methods provided real insights into the truth. In the remainder of this paper, we use the notation [n] to denote the n-element set $\{1,2,\ldots ,n\}$. (No order is implied on [n], except for the natural order on positive integers.)

Theorem 1.

Let P = (X,P) be a poset with |X| = n. Then X contains a fiber of cardinality at most 4n∕5. Consequently, fib (n) ≤ 4n∕5.

Proof.

Let C ⊆ X be a maximum chain. Then X − C is a fiber. So we may assume that | C |  < n ∕ 5. Label the points of C as $x_{1} < x_{2} < \cdots < x_{t}$, where $t = \vert C\vert < n/5$. Next we define two different partitions of X − C. First, for each i ∈ [t], set $U_{i} =\{ x \in X - C$: i is the least integer for which x ∥ x _i }. Then set $D_{i} =\{ x \in X - C$: i is the largest integer for which x ∥ x _i }.

Then for each subset S ⊆ [t − 1], define

$$\displaystyle{B(S) = C \cup (\cup \{D_{i} : i \in S\}) \cup (\cup \{U_{i+1} : i\notin S\})}$$

Note that for each i ∈ [t − 1], the maximality of C implies that $D_{i} \cap U_{i+1} = \varnothing$.

Claim 1.

For every subset S ⊆ [t − 1], B(S) is a fiber.

Proof.

Let S ⊆ [t − 1] and let A be a non-trivial maximal antichain. We show that $A \cap B(S)\neq \varnothing$. This intersection is nonempty if $A \cap C\neq \varnothing$, so we may assume that $A \cap C = \varnothing$. Now the fact that C is a maximal chain implies that every point of C is comparable with one or more points of A. However, no point of C can be greater than one point of A and less than another point of A. Also, x ₁ can only be less than points in A, and x _t can only be greater than points in A. It follows that t ≥ 2 and that there is an integer i ∈ [t − 1] and points a, a′ ∈ A for which x _i  < a in P and x _i + 1 > a′ in P. Clearly, a′ ∈ D _i and a ∈ U _i + 1. If i ∈ S, then D _i  ⊂ B(S), and if i∉S, then U _i + 1 ⊂ B(S). In either case, we conclude that $A \cap B(S)\neq \varnothing$.

Claim 2.

The expected cardinality of B(S) with all subsets S ⊆ [t − 1] equally likely is $t + 3(n - t)/4$.

Proof.

Note that C ⊆ B(S), for all S. For each element x ∈ X − C, let i and j be the unique integers for which x ∈ D _i and x ∈ U _j . Then j≠i + 1. It follows that the probability that x belongs to B(S) is exactly 3/4.

To complete the proof of the theorem, we note that there is some S ⊆ [t − 1] for which the fiber B(S) has at most $t + 3(n - t)/4$ points. However, t < n ∕ 5 implies that $t + 3(n - t)/4 < 4n/5$.

The preceding theorem remains an interesting (although admittedly elementary) illustration of applying random methods to general partially ordered sets. Characteristically, it shows that an n-point poset has a fiber containing at most 4n ∕ 5 points without actually producing the fiber. Furthermore, this is also an instance in which the constant provided by random methods can be improved by another approach.

The following result is due to Duffus, Kierstead and Trotter [5].

Theorem 2 (Duffus, Kierstead and Trotter). 

Let P = (X,P) be a poset and let $\mathcal{H}$ be the hypergraph of non-trivial maximal antichains of P . Then the chromatic number of $\mathcal{H}$ is at most 3.

Theorem 2 shows that fib(n) ≤ 2n ∕ 3, since whenever $X = B_{1} \cup B_{2} \cup B_{3}$ is a 3-coloring of the hypergraph $\mathcal{H}$ of non-trivial maximal antichains, then the union of any two of $\{B_{1},B_{2},B_{3}\}$ is a fiber. Quite recently, Lone [17] has obtained the following interesting result providing a better upper bound for posets with small width.

Theorem 3 (Lonc). 

Let P = (X,P) be a poset of width 3 and let |X| = n. P has a fiber of cardinality at most 11n∕18.

I am still tempted to assert that $\lim _{n\rightarrow \infty }\mathrm{fib}(n)/n = 2/3$.

3 Dimension Theory

When P = (X, P) is a poset, a linear order L on X is called a linear extension of P when x < y in L for all x, y ∈ X with x < y in P. A set $\mathcal{R}$ of linear extensions of P is called a realizer of P when $P = \cap \mathcal{R}$, i.e., for all x, y in X, x < y in P if and only if x < y in L, for every $L \in \mathcal{R}$. The minimum cardinality of a realizer of P is called the dimension of P and is denoted dim(P).

It is useful to have a simple test to determine whether a family of linear extensions of P is actually a realizer. The first such test is just a reformulation of the definition. Let inc(P) = inc(X, P) denote the set of all incomparable pairs in P. Then a family $\mathcal{R}$ of linear extensions of P is a realizer of P if and only if for every (x, y) ∈ inc(X, P), there exist distinct linear extensions L, $L^{\prime} \in \mathcal{R}$ so that x > y in L and y > x in L′.

Here is a more useful test. Call a pair (x, y) ∈ X ×X a critical pair if:

• 1. x ∥ y in P;

• 2. z < x in P implies z < y in P, for all z ∈ X; and

• 3. w > y in P implies w > x in P, for all w ∈ X.

The set of all critical pairs of P is denoted crit(P) or crit(X, P). Then it is easy to see that a family $\mathcal{R}$ of linear extensions of P is a realizer of P if and only if for every critical pair (x, y), there is some $L \in \mathcal{R}$ with x > y in L. We say that a linear order L on Xreverses (x, y) if x > y in L. So the dimension of a poset is just the minimum number of linear extensions required to reverse all critical pairs.

For each n ≥ 3, let S _n denote the height 2 poset with n minimal elements $a_{1},a_{2},\ldots ,a_{n}$, n maximal elements $b_{1},b_{2},\ldots ,b_{n}$ and a _i  < b _j , for i, j ∈ [n] and j≠i. The poset S _n is called the standard example of an n-dimensional poset. Note that dim(S _n ) is at most n, since $\mathrm{crit}(\mathbf{S}_{n}) =\{ (a_{i},b_{i}) : i \in [n]\}$ and n linear extensions suffice to reverse the n critical pairs in crit(S _n ). On the other hand, dim(S _n ) ≥ n, since no linear extension can reverse more than one critical pair.

4 The Dimension of Subposets of the Subset Lattice

For integers k, r and n with 1 ≤ k < r < n, let P(k, r; n) denote the poset consisting all k-element and all r-element subsets of $\{1,2,\ldots ,n\}$ partially ordered by inclusion. For simplicity, we use dim(k, r; n) to denote the dimension of P(k, r; n).

Historically, most researchers have concentrated on the case k = 1. In a classic 1950 paper in dimension theory, Dushnik [7] gave an exact formula for dim(1, r; n), when $r \geq 2\sqrt{n}$.

Theorem 4 (Dushnik). 

Let n, r and j be positive integers with n ≥ 4 and $2\sqrt{n} - 2 \leq r < n - 1$ . If j is the unique integer with $2 \leq j \leq \sqrt{n}$ for which

$$\displaystyle{\left \lfloor \frac{n - 2j + {j}^{2}} {j} \right \rfloor \leq k < \left \lfloor \frac{n - 2(j - 1) + {(j - 1)}^{2}} {j - 1} \right \rfloor ,}$$

then $\dim (1,r;n) = n - i + 1$ .

No general formula for dim(1, r; n) is known when r is relatively small in comparison to n, although some surprisingly tight estimates have been found. Here is a very brief overview of this work, beginning with an elementary reformulation of the problem. When L is a linear order on X, S ⊂ X and x ∈ X − S, we say x > S in L when x > s in L, for every s ∈ S.

Proposition 1.

dim (1,r;n) is the least t so that there exist t linear orders $L_{1},L_{2},\ldots L_{t}$ of [n] so that for every r-element subset S ⊂ [n] and every x ∈ [n] − S, there is some i ∈ [t] for which x > S in L _i .

Spencer [20] used this proposition to estimate dim(1, 2; n). First, he noted that by the Erdős-Szekeres theorem, if $n > {2}^{{2}^{t} }$ and $\mathcal{R}$ is any set of t linear orders on [n], then there exists a 3-element set $\{x,y,z\} \subset [n]$ so that for all $L \in \mathcal{R}$, either x < y < z in L or x > y > z in L. Thus dim(1, 2; n) > t when $n > {2}^{{2}^{t} }$. On the other hand, if $n \leq {2}^{{2}^{t} }$, then there exists a family $\mathcal{R}$ of t linear orders on [n] so that for every 3-element subset S ⊂ [n] and every x ∈ S, there exists some $L \in \mathcal{R}$ so that either $x < S -\{ x\}$ in L or $x > S -\{ x\}$ in L. Then let S be the family of 2t linear orders on X determined by adding to $\mathcal{R}$ the duals of the linear orders in $\mathcal{R}$. Clearly, the 2t linear orders in S satisfy the requirements of Proposition 1 when r = 2, and we conclude:

Theorem 5 (Spencer). 

For all n ≥ 4,

$$\displaystyle{\lg \,\lg \,n <\dim (1,2;n) \leq 2\,\lg \,\lg \,n.}$$

Spencer [20] then proceeded to determine a more accurate upper bound for dim(1, 2; n) using a technique applicable to larger values of r. Let t be a positive integer, and let $\mathcal{F}$ be a family of subsets of [t]. Then let r be an integer with 1 ≤ r ≤ t. We say $\mathcal{F}$ is r-scrambling if $\vert \mathcal{F}\vert \geq r$ and for every sequence $(A_{1},A_{2},\ldots ,A_{r})$ of r distinct sets from $\mathcal{F}$ and for every subset B ⊆ [r], there is an element α ∈ [t] so that α ∈ A _β if and only if β ∈ B. We let M(r, t) denote the maximum size of a r-scrambling family of subsets of [t]. Spencer then applied the Erdős/Ko/Rado theorem to provide a precise answer for the size of M(2, t).

Theorem 6 (Spencer). 

$M(2,t) = \left ({ t-1 \atop \lfloor \frac{t-2} {2} \rfloor } \right )$ , for all t ≥ 4.

As a consequence, Spencer observed that

$$\displaystyle{\lg \,\lg \,n <\dim (1,2;n) \leq \lg \,\lg \, n + (\tfrac{1} {2} + o(1))\lg \,\lg \,\lg \,n.}$$

Almost 20 years later, Füredi, Hajnal, Rödl and Trotter [13] were able to show that the upper bound in this inequality is tight, i.e.,

$$\displaystyle{\dim (1,2;n) =\lg \,\lg \, n + (\tfrac{1} {2} + o(1))\lg \,\lg \,\lg \,n.}$$

For larger values of r, Spencer used random methods to produce the following bound.

Theorem 7 (Spencer). 

For every r ≥ 2, there exists a constant c = c _r > 1 so that M(r,t) > c ^t .

Proof.

Let p be a positive integer and consider the set of all sequences of length p whose elements are subsets of [t]. There are 2^pt such sequences. The number of such sequences which fail to be r-scrambling is easily seen to be at most

$$\displaystyle{\left ({ p \atop r} \right ){2}^{r}{({2}^{r} - 1)}^{t}{2}^{(p-r)t}.}$$

So at least one of these sequences is a r-scrambling family of subsets of [t] provided $\left ({ p \atop r} \right ){2}^{r}{({2}^{r} - 1)}^{t}{2}^{(p-r)t} < {2}^{pt}$. Clearly this inequality holds for p > c ^t where $c = c_{r} \sim {e}^{ \frac{1} {r{2}^{r}} }$ is a constant larger than 1.

Here’s how the concept of scrambling families is used in provide upper bounds for dim(q, r; n).

Theorem 8 (Spencer). 

If p = M(r,t) and n = 2 ^p , then dim (1,r;n) ≤ t.

Proof.

Let $\mathcal{F}$ be an r-scrambling family of subsets of [t], say $\mathcal{F} =\{ A_{1},A_{2},\ldots ,A_{p}\}$ where p = M(r, t). Then set n = 2^p and let $Q_{1},Q_{2},\ldots ,Q_{n}$ be the subsets of [p]. For each α ∈ [t], define a linear order L _α on the set [n] by the following rules. Let x and y be distinct integers from [n] and let $u =\min ((Q_{x} - Q_{y}) \cup (Q_{y} - Q_{x}))$. Set x > y in L _α if either

• 1. α  ∈ A _u and $u\in Q_{x} - Q_{y}$, or

• 2. α ∉A _u and $u\in Q_{y} - Q_{x}$.

It is not immediately clear why L _α is a linear order on [n] for each α ∈ [t], but it is easy to check that this is so. Now let S be an r-element subset of [n] and let x ∈ [n] − S. We must show that x > S in L _α for some α ∈ [t]. For each y ∈ S, let $u_{y} =\min ((Q_{x} - Q_{y}) \cup (Q_{y} \cup Q_{x}))$ and then consider the family $\{A_{u_{y}} : y \in S\}$. Since ℱ is a r-scrambling family of subsets of [t], there exists some α ∈ [t] such that $\alpha \in A_{u_{y}}$ if and only if u _y  ∈ Q _x . It follows from the definition of L _α that x > S in L _α .

By paying just a bit of attention to constants, the preceding results of Spencer actually yield the following upper bound on dim(1, r; n).

Theorem 9 (Spencer). 

For all r ≥ 2, $\dim (1,r;n) \leq (1 + o(1))\frac{1} {\lg \,e}r{2}^{r}\,\lg \,\lg \,n$ .

Of course, this bound is only meaningful if r is relatively small in comparison to n, but in this range, it is surprisingly tight. The following lower bound is a quite recent result due to Kierstead.

Theorem 10 (Kierstead). 

If 2 ≤ r ≤ lg  lg  n − lg  lg  lg  n, then

$$\displaystyle{\frac{{(r + 2 -\lg \,\lg \, n +\lg \,\lg \lg \, n)}^{2}\lg \,n} {32\ \lg (r + 2 -\lg \,\lg \, n +\lg \,\lg \,\lg \, n)} \leq \dim (1,r;n).}$$

We will return to the issue of estimating dim(1, r; n) in the next section.

5 The Dimension of Posets of Bounded Degree

Given a poset P = (X, P) and a point x ∈ X, define the degree of x in P, denoted deg _P (x), as the number of points in X which are comparable to x, This is just the degree of the vertex x in the associated comparability graph. Then define $\Delta (\mathbf{P})$ as the maximum degree of P. Finally, define Dim(k) as the maximum dimension of a poset P with $\Delta (\mathbf{P}) \leq k$. Rödl and Trotter were the first to prove that Dim(k) is well defined. Their argument showed that Dim(k) ≤ 2k ² + 2. It is now possible to present a very short argument for this result by first developing the following idea due to Füredi and Kahn [12].

For a poset P = (X, P) and a point x ∈ X, let $U(x) =\{ y \in X : y > x$ in $P\}$ and let $U[x] = U(x) \cup \{ x\}$. Dually, let $D(x) =\{ y \in X : y < x$ in $P\}$ and $D[x] = D(x) \cup \{ x\}$. The following proposition admits an elementary proof. In fact, something more can be said, and we will comment on this in the next section.

Proposition 2 (Füredi and Kahn). 

Let P = (X,P) be a poset and let L be any linear order on X. Then there exist a linear extension L′ of P so that if (x,y) is a critical pair and x > D[y] in L, then x > y in L′, so that x > D[y] in L′.

Theorem 11 (Rödl and Trotter). 

If P = (X,P) is a poset with $\Delta (\mathbf{P}) \leq k$ , then dim ( P ) ≤ 2k ² + 2.

Proof.

Define a graph G = (X, E) as follows. The vertex set X is the ground set of P. The edge set E contains those two element subsets $\{x,y\}$ for which U[x] ∩ U[y]≠∅. Clearly, the maximum degree of a vertex in G is at most k ². Therefore, the chromatic number of G is at most k ² + 1. Let $t = {k}^{2} + 1$ and let $X = X_{1} \cup X_{2} \cup \ldots \cup X_{t}$ be a partition of X into subsets which are independent in G. Then for each i ∈ [t], let L _i be any linear order on X with X _i  > X − X _i in L _i . Finally, define L _t + i to be any linear order on X so that:

• 1. X _i  > X − X _i in L _t + i , and

• 2. The restriction of L _t + i to X _i is the dual of the restriction of L _i to X _i .

We claim that for every critical pair (x, y) ∈ crit(P), if x ∈ X _i , then either x > D[y] in L _i or x > D[y] in L _t + i . This claim follows easily from the observation that any two points of D[y] are adjacent in G so that | D[y] ∩ X _i  | ≤  1.

Füredi and Kahn [12] made a dramatic improvement in the upper bound for in Dim(k) by applying the Lovász Local Lemma [9]. We sketch their argument which begins with an application of random methods to provide an upper bound for dim(1, r; n). In this sketch, we make no attempt to provide the best possible constants.

Theorem 12 (Füredi and Kahn). 

Let r and n be integers with 1 < r < n. If t is an integer such that

$$\displaystyle{ n\left ({ n - 1 \atop r} \right ){( \frac{r} {r + 1})}^{t} < 1, }$$

(1)

then dim (1,r;n) ≤ t. In particular, $\dim (1,r;n) \leq r(r + 1)\log (n/r)$ .

Proof.

Let t be an integer satisfying the inequality given in the statement of the theorem. Then let $\{L_{i} : i \in [t]\}$ be a sequence of t random linear orders on X. The expected number of pairs (x, S) where S is an r-element subset of [n], x ∈ [n] − S and there is no i ∈ [t] for which x > S in L _i is exactly what the left hand side of this inequality is calculating. It follows that this quantity is less than one, so the probability that there are no such pairs is positive. This shows that dim(1, r; n) ≤ t. The estimate $\dim (1,r;n) \leq r(r + 1)\log (n/r)$ follows easily.

Theorem 13 (Füredi and Kahn). 

If P = (X,P) is a poset for which $\Delta (\mathbf{P}) \leq k$ , then dim ( P ) ≤ 100k log² k, i.e., Dim (k) ≤ 100k log² k.

Proof.

The inequality dim(P) ≤ 100k log² k follows from the preceding theorem if k ≤ 1, 000, so we may assume that k > 1, 000. Set $m = \lceil k/\log \,k\rceil$ and r = ⌈9 log k⌉. Using the Lovász Local Lemma, we see that there exists a partition $X = Y _{1} \cup Y _{2}\ldots \cup Y _{m}$, with | D[x] ∩ Y _i  | ≤ r, for every x ∈ X. Now fix i ∈ [m], let $q = rk + 1$ and let s = dim(1, r; q). We construct a family $\mathcal{R}_{\rangle } =\{ \mathcal{L}_{\rangle ,\vert } : \vert \in [\in \int ]\}$ as follows.

Let G be the graph on X defined in the proof of Theorem 11. Then let G _i be the subgraph induced by Y _i . Now it is easy to see that any point of Y _i is adjacent to at most rk other points in Y _i in the graph G _i . It follows that the chromatic number of G _i is at most rk + 1. Let $Y _{i} = Y _{i,1} \cup \ldots \cup Y _{i,q}$ be a partition into subsets each of which is independent in G _i . Then let $\mathcal{R} =\{ \mathcal{M}_{\vert } : \vert \in [\int ]\}$ be a family of linear orders of [q] so that for every r-element subset S ⊂ [q] and every x ∈ [q] − S, there is some j ∈ [s] for which x > S in M _j .

Then for each j ∈ [s], define L _i, j as any linear order for which:

• 1. Y _i  > X − Y _i in L _i, j and

• 2. If a < b in M _j , then Y _i, a  < Y _i, b in L _i, j .

Finally, for each j ∈ [s], define L _{i, s + j} as any linear order for which:

• 1. Y _i  > X − Y _i in L _{i, s + j} ,

• 2. If a < b in M _j , then Y _i, a  < Y _i, b in L _i, j and

• 3. If a ∈ [q], then the restriction of L _{i, s + j} to Y _i, a is the dual of the restriction of L _i, j to Y _i, a .

Next we claim that if (x, y) is a critical pair and x ∈ Y _i , then there is some j ∈ [2s] so that x > D[y] in L(i, j). To see this observe that any two points in D[y] are adjacent in G so at most r points in D[y] belong to Y _i , and all points of D[y] ∩ Y _i belong to distinct subsets in the partition of Y _i into independent subsets. Let x ∈ Y _i, j0. Then there exists some j ∈ [s] so that j ₀ > j in M _j whenever j≠j ₀ and D[y] ∩ Y _i, j ≠∅. It follows that either x > D[y] in L(i, j) or x > D[y] in L(i, s + j).

Finally, we note that $s =\dim (1,r;q) \leq r(r + 1)\log (q/r)$, so that dim(P) ≤ 100k log² k as claimed.

There are two fundamentally important problems which leap out from the preceding inequality limiting the dimension of posets of bounded degree, beginning with the obvious question: Is the inequality Dim(k) = O(k log² k) best possible? However, the details of the proof also suggest that the inequality could be improved if one could provide a better upper bound than dim(1, log k; k) = O(log³ k). Unfortunately, the second approach will not yield much as Kierstead [15] has recently provided the following lower bound.

Theorem 14 (Kierstead). 

If $\lg \,\lg \,n -\lg \,\lg \,\lg \, n \leq r \leq {2}^{{\lg }^{1/2}n }$ , then

$$\displaystyle{ \frac{{(r + 2 -\lg \,\lg \, n +\lg \,\lg \,\lg \,\lg \, n)}^{2}\lg \,n} {32\lg (r + 2 -\lg \,\lg \, n +\lg \,\lg \,\lg \, n)} \leq \dim (1,r;n) \leq { \frac{2{k{}^{2}\lg }^{2}n} {\lg }^{2}k} . }$$

(2)

As a consequence, it follows that $\dim (1,\log \,k;k) = \Omega {(\log }^{3}k/\log \,\log \,k)$. So the remaining challenge is to provide better lower bounds on Dim(k). Random methods seem to be our best hope. Here is a sketch of the technique used by Erdős, Kierstead and Trotter [8] to show that $\mathrm{Dim}(k) = \Omega (k\,\log \,k)$.

For a fixed positive integer n, consider a random poset P _n having n minimal elements $a_{1},a_{2},\ldots ,a_{n}$ and n maximal elements $b_{1},b_{2},\ldots ,b_{n}$. The order relation is defined by setting a _i  < b _j with probability p = p(n); also, events corresponding to distinct min-max pairs are independent.

Erdős, Kierstead and Trotter then determine estimates for the expected value of the dimension of the resulting random poset. The arguments are far too complex to be conveniently summarized here, as they make non-trivial use of correlation inequalities. However, the following theorem summarizes the lower bounds obtained in [8].

Theorem 15 (Erdős, Kierstead and Trotter). 

• 1. For every ε > 0, there exists δ > 0 so that if

  $${\displaystyle\frac{{\log }^{1+\epsilon }n} {n} < p \leq \frac{1} {\log \,n},}$$

  then

  $$\displaystyle{\dim (\mathbf{P}) >\delta pn\,\log \,pn,\mbox{ for almost all }\mathbf{P}.}$$

• 2. For every ε > 0, there exist δ ,c > 0 so that if

  $$\displaystyle{\frac{1} {\log \,n} \leq p < 1 - {n}^{-1+\epsilon },}$$

  then

  $$\displaystyle{\dim (\mathbf{P}) >\max \{\delta n,n -\frac{\mathit{cn}} {p\,\log \,n}\},\mbox{ for almost all }\mathbf{P}.}$$

The following result is then an easy corollary.

Corollary 1 (Erdős, Kierstead and Trotter). 

For every ε > 0, there exists δ > 0 so that if

$$\displaystyle{{n}^{-1+\epsilon } < p \leq \frac{1} {\log \,n},}$$

then

$$\displaystyle{\dim (\mathbf{P}) >\delta \Delta (\mathbf{P})\log \,n,for\ almost\ all\ \mathbf{P}.}$$

Summarizing, we now know that

$$\displaystyle{ \Omega (k\,\log \,k) = D(k) = O({k\,\log }^{2}k). }$$

(3)

It is the author’s opinion that the upper bound is more likely to be correct and that the proof of this assertion will come from investigating the dimension of a slightly different model of random height 2 posets. For integers n and k with k large but much smaller than n, we consider a poset with n minimal points and n maximal points. However, the comparabilities come from taking k random matchings.

The techniques used by Erdős, Kierstead and Trotter in [8] break down when $p = o(\log \,n/n)$. But this is just the point at which we can no longer guarantee that the maximum degree is O(pn).

6 Fractional Dimension and Ramsey Theory for Probability Spaces

In many instances, it is useful to consider a fractional version of an integer valued combinatorial parameter, as in many cases, the resulting LP relaxation sheds light on the original problem. In [3], Brightwell and Scheinerman proposed to investigate fractional dimension for posets. This concept has already produced some interesting results, and many appealing questions have been raised. Here’s a brief sketch of some questions with immediate connections to random methods.

Let P = (X, P) be a poset and let $\mathcal{F} =\{ \mathcal{M}_{\infty },\ldots ,\mathcal{M}_{\sqcup }\}$ be a multiset of linear extensions of P. Brightwell and Scheinerman [3] call $\mathcal{F}$ a k-fold realizer of P if for each incomparable pair (x, y), there are at least k linear extensions in $\mathcal{F}$ which reverse the pair (x, y), i.e., $\vert \{i : 1 \leq i \leq t,x > y$ in $M_{i}\}\vert \geq k$, The fractional dimension of P, denoted by fdim(P), is then defined as the least real number q ≥ 1 for which there exists a k-fold realizer $\mathcal{F} =\{ M_{1},\ldots ,M_{t}\}$ of P so that $k/t \geq 1/q$ (it is easily verified that the least upper bound of such real numbers q is indeed attained and is a rational number). Using this terminology, the dimension of P is just the least t for which there exists a 1-fold realizer of P. It follows immediately that fdim(P) ≤ dim(P), for every poset P.

Note that the standard example of an n-dimensional poset also has fractional dimension n. Brightwell and Scheinerman [3] proved that if P is a poset and | D(x) | ≤ k, for all x ∈ X, then fdim(P) ≤ k + 2. They conjectured that this inequality could be improved to fdim(P) ≤ k + 1. This was proved by Felsner and Trotter [10], and the argument yielded a much stronger conclusion, a result with much the same flavor as Brooks’ theorem for graphs.

Theorem 16 (Felsner and Trotter). 

Let k be a positive integer, and let P be any poset with |D(x)|≤ k, for all x ∈ X. Then fdim ( P ) ≤ k + 1. Furthermore, if k ≥ 2, then fdim ( P ) < k + 1 unless one of the components of P is isomorphic to S _k+1 , the standard example of a poset of dimension k + 1.

We do not discuss the proof of this result here except to comment that it requires a strengthening of Proposition 2, and to note that it implies that the fractional dimension of the poset P(1, r; n) is r + 1. Thus a poset can have large dimension and small fractional dimension. However, there is one elementary bound which limits dimension in terms of fractional dimension.

Theorem 17.

If P = (X,P) is a poset with |X| = n and fdim ( P ) = q, then dim ( P ) ≤ (2 + o(1))q log  n.

Proof.

Let $\mathcal{F}$ be a multi-realizer of P so that $\mathrm{Prob}_{\mathcal{F}}[x > y] \geq 1/q$, for every critical pair (x, y) ∈ crit(P). Then take t to be any integer for which

$$\displaystyle{n(n - 1){(1 - 1/q)}^{t} < 1.}$$

Then let $\{L_{1},\ldots ,L_{t}\}$ be a sequence of length t in which the linear extensions in $\mathcal{F}$ are equally likely to be chosen. Then the expected number of critical pairs which are not reversed is less than one, so the probability that we have a realizer of cardinality t is positive.

Felsner and Trotter [10] derive several other inequalities for fractional dimension, and these lead to some challenging problems as to the relative tightness of inequalities similar to the one given in Theorem 16. However, the subject of fractional dimension has produced a number of challenging problems which are certain to require random methods in their solutions. Here is two such problems, one of which has recently been solved.

A poset P = (X, P) is called an interval order if there exists a family $\{[a_{x},b_{x}] : x \in X\}$ of non-empty closed intervals of $\mathbb{R}$ so that x < y in P if and only if b _x  < a _y in $\mathbb{R}$. Fishburn [11] showed that a poset is an interval order if and only if it does not contain 2 + 2 as a subposet. The interval order I _n consisting of all intervals with integer endpoints from $\{1,2,\ldots ,n\}$ is called the canonical interval order.

Although posets of height 2 can have arbitrarily large dimension, this is not true for interval orders. For these posets, large height is a prerequisite for large dimension.

Theorem 18 (Füredi, Hajnal, Rödl and Trotter). 

If P = (X,P) is an interval order of height n, then

$$\displaystyle{ \dim (\mathbf{P}) \leq \lg \,\lg \, n + (1/2 + o(1))\lg \,\lg \,\lg \,n. }$$

(4)

The inequality in the preceding theorem is best possible.

Theorem 19 (Füredi, Hajnal, Rödl and Trotter). 

The dimension of the canonical interval order satisfies

$$\displaystyle{ \dim (\mathbf{I}_{n}) =\lg \,\lg \, n + (1/2 + o(1))\lg \,\lg \,\lg \,n. }$$

(5)

Although interval orders may have large dimension, they have bounded fractional dimension. Brightwell and Scheinerman [3] proved that the dimension of any finite interval order is less than 4, and they conjectured that for every ε > 0, there exists an interval order with dimension greater than 4 − ε. We believe that this conjecture is correct, but confess that our intuition is not really tested. For example, no interval order is known to have fractional dimension greater than 3.

Motivated by the preceding inequalities and the known bounds on the dimension and fractional dimension of interval orders and the posets P(1, r; n), Brightwell asked whether there exists a function $f : \mathbb{Q} \rightarrow \mathbb{R}$ so that if P = (X, P) is a poset with | X |  = n and fdim(P) = q, then dim(P) ≤ f(q)lg lg n. If such a function exists, then the family P(1, r; n) shows that we would need to have $f(q) = \Omega ({2}^{q})$.

But we will show that there is no such function. The argument requires some additional notation and terminology. Fix integers n and k with 1 ≤ k < n. We call an ordered pair (A, B) of k-element sets a (k, n)-shift pair if there exists a (k + 1)-element subset $C =\{ i_{1} < i_{2} < \cdots < i_{k+1} \subseteq \{ 1,2,\ldots ,n\}$ so that $A =\{ i_{1},i_{2},\ldots ,i_{k}\}$ and $B =\{ i_{2},i_{3},\ldots ,i_{k+1}$. We then define the (k, n)-shift graphS(k, n) as the graph whose vertex set consists of all k-element subsets of $\{1,2,\ldots ,n\}$ with a k-element set A adjacent to a k-element set B exactly when (A, B) is a (k, n)-shift pair. Note that the (1, n) shift graph S(1, n) is just a complete graph. It is customary to call a (2, n)-shift graph just a shift graph; similarly, a (3, n)-shift graph is called a double shift graph. The formula for the chromatic number of the (2, n)-shift graph S(2, n) is a folklore result of graph theory: χ(S(2, n)) = ⌈lg n⌉. Several researchers in graph theory have told me that this result is due to Andras Hajnal, but Andras says that it is not. In any case, it is an easy exercise.

The following construction exploits the properties of the shift graph to provide a negative answer for Brightwell’s question.

Theorem 20.

For every m ≥ 3, there exists a poset P = (X,P) so that

• 1. |X| = m ²;

• 2. dim (X,P) ≥ lg  m; and

• 3. fdim (X,P) ≤ 4.

Proof.

The poset P = (X, P) is constructed as follows. Set $X =\{ x(i,j) : 1 \leq i,j \leq m\}$, so that | X |  = m ². The partial order P is defined by first defining $x(i,j_{1}) < x(i,j_{2})$ in P, for each i ∈ [m] whenever $1 \leq j_{1} < j_{2} \leq m$. Furthermore, for each i ∈ [m], $x(i_{1},j_{1}) < x(i_{2},j_{2})$ in P if and only if $(i_{2} - i_{1}) + (j_{2} - j_{1}) > m$.

We now show that dim(X, P) ≥ lg m. Note first that for each i, j with 1 ≤ i < j ≤ m, x(i, j − i) ∥ x(j, m). Let dim(X, P) = t, and let $\mathcal{R} =\{ \mathcal{L}_{\infty },\mathcal{L}_{\in },\ldots ,\mathcal{L}_{\sqcup }\}$ be a realizer of P. For each i, j with 1 ≤ i < j ≤ m, choose an integer $\phi (\{i,j\}) =\alpha \in \{ 1,2,\ldots ,t\}$ so that x(i, j − i) > x(j, m) in L _α . We claim that ϕ is a proper coloring of the (2, m) shift graph S(1, m) using t colors, which requires that $\dim (X,P) = t \geq \chi (\mathbf{S}(2,m) = \lceil \lg \,m\rceil$. To see that ϕ is a proper coloring, let i, j and k be integers with 1 ≤ i < j < k ≤ m, let $\phi (\{i,j\}) =\alpha$ and let $\phi (\{j,k\}) =\beta$. If α = β, then x(i, j − i) > x(j, m) in L _α and x(j, k − j) > x(k, m) in L _α . Also, x(j, m) > x(j, k − j) in P. However, since $(k - i) + (m - j + i) > m$, it follows that x(k, m) > x(i, j − i) in P, so that x(k, m) > x(i, j − i) in L _α . Thus,

$$\displaystyle{ x(i,j - i) > x(j,m) > x(j,k - j) > x(k,m) > x(i,j - i)\mbox{ in }P }$$

(6)

The inequalities in equation 6 cannot all be true. The contradiction shows that ϕ is a proper coloring of the shift graph S(2, m) as claimed. In turn, this shows that dim(X, P) ≥ ⌈lg m⌉.

Finally, we show that fdim(X, P) ≤ 4. For each element x ∈ X, let p ₁ and p ₂ be the natural projection maps defined by p(x) = i and p ₂(x) = j when x = x(i, j). Next, we claim that for each subset A ⊂ [m], there exists a linear extension L(A) of P so that x > y in L(A) if:

• 1. x ∥ y in P;

• 2. p ₁(x) ∈ A and p ₁(y)∉A.

To show that such linear extensions exist, we use the alternating cycle test (see Chap. 2 of [23]). Let A ⊆ [m], and let $S(A) =\{ (x,y) \in X \times X : x \parallel y$ in P, p ₁(x) ∈ A and p ₁(y)∉A}. Now suppose that $\{(u_{k},v_{k}) : 1 \leq k \leq p\} \subseteq S(A)$ is an alternating cycle of length p, i.e., u _k  ∥ v _k and u _k  ≤ v _k + 1 in P, for all k ∈ [p] (subscripts are interpreted cyclically). Let k ∈ [m]. Then p ₁(u _k ) ∈ A and p ₁(v _k + 1)∉A. It follows that u _k  < v _k + 1 in P, for each k ∈ [p]. It follows that $p_{1}(v_{k+1} - p_{1}(u_{k}) + p_{2}(v_{k+1}) - p_{2}(u_{k}) > m$. Also, we know that $m \geq p_{1}(v_{k}) - p_{1}(u_{k}) + p_{2}(v_{k}) - p_{2}(u_{k})$. Thus $p_{1}(v_{k+1}) + p_{2}(v_{k+1}) > p_{1}(v_{k}) + p_{2}(v_{k})$. Clearly, this last inequality cannot hold for all k ∈ [p]. The contradiction shows that S(A) cannot contain any alternating cycles. Thus the desired linear extension L(A) exists.

Finally, we note that if we take $\mathcal{F} =\{ \mathcal{L}(A) : \mathcal{A}\subseteq [\Updownarrow ]\}$ and set $s = \vert \mathcal{F}\vert$, then x > y in at least s ∕ 4 of the linear extensions in $\mathcal{F}$, whenever x ∥ y in P. To see this, observe that there are exactly 2^s ∕ 4 subsets of [m] which contain p ₁(x) but do not contain p ₁(y). This shows that fdim(X, P) ≤ 4 as claimed. It also completes the proof of the theorem.

Now we turn our attention to the double shift graph. If P = (X, P) is a poset, a subset D ⊆ X is called a down set, or an order ideal, if x ≤ y in P and y ∈ D always imply that x ∈ P. The following result appears in [13] but may have been known to other researchers in the area.

Theorem 21.

Let n be a positive integer. Then the chromatic number of the double shift graph S (3,n) is the least t so that there are at least n down sets in the subset lattice 2 ^t .

The problem of counting the number of down sets in the subset lattice 2 ^t is a classic problem and is traditionally called Dedekind’s problem. Although no closed form expression is known, relatively tight asymptotic formulas have been given. For our purposes, the estimate provided by Kleitman and Markovsky [16] suffices. Theorem 21, coupled with the estimates from [16] permit the following surprisingly accurate estimate on the chromatic number χ(S(3, n)) of the double shift graph [13].

Theorem 22 (Füredi, Hajnal, Rödl and Trotter). 

$$\displaystyle{\chi (\mathbf{S}(3,n)) =\lg \,\lg \, n + (1/2 + o(1))\lg \,\lg \,\lg \,n.}$$

Now that we have introduced the double shift graph, the following elementary observation can be made [13].

Proposition 3.

For each n ≥ 3, dim (1,2;n) ≥χ( S (3,n)), and dim ( I _n ) ≥χ( S (3,n)).

Although the original intent was to investigate questions involving the fractional dimension of posets, Trotter and Winkler [27] began to attack a Ramsey theoretic problem for probability spaces which seems to have broader implications. Fix an integer k ≥ 1, and let n ≥ k + 1. Now suppose that $\Omega$ is a probability space containing an event E _s for every k-element subset $S \subset \{ 1,2,\ldots ,n\}$. We abuse terminology slightly and use the notation Prob(S) rather than Prob(E _S ).

Now let $f(\Omega )$ denote the minimum value of $\mathrm{Prob}(A\overline{B})$, taken over all (k, n)-shift pairs (A, B). Note that we are evaluating the probability that A is true and B is false. Then let f(n, k) denote the maximum value of $f(\Omega )$ and let f(k) denote the limit of f(n, k) as n tends to infinity.

Even the case k = 1 is non-trivial, as it takes some work to show that $f(1) = 1/4$. However, there is a natural interpretation of this result. Given a sufficiently long sequence of events, it is inescapable that there are two events, A and B with A occurring before B in the sequence, so that

$$\displaystyle{\mathrm{Prob}(A\overline{B}) < \frac{1} {4} +\epsilon .}$$

The $\frac{1} {4}$ term in this inequality represents coin flips. The ε is present because, for finite n, we can always do slightly better than tossing a fair coin.

For k = 2, Trotter and Winkler [27] show that $f(2) = 1/3$. Note that this is just the fractional chromatic number of the double shift graph. This result is also natural and comes from taking a random linear order L on $\{1,2,\ldots ,n\}$ and then saying that a 2-element set $\{i,j\}$ is true if i < j in L. Trotter and Winkler conjecture that $f(3) = 3/8$, $f(4) = 2/5$, and are able to prove that $\lim _{k\rightarrow \infty }f(k) = 1/2$. They originally conjectured that $f(k) = k/(2k + 2)$, but they have since been able to show that $f(5) \geq \frac{27} {64}$ which is larger than $\frac{5} {12}$.

As an added bonus to this line of research, we are beginning to ask natural (and I suspect quite important) questions about patterns appearing in probability spaces.