Epistemologías Emergentes-Epistemologías del Sur

(c)^∗ Prove that every countable linearly ordered set admits an embedding into (Q,≤).

7. ^∗Prove that every subset of the poset Bn has both supremum and infimum (see Exercise 2.2.9 for the definitions).

8. Count the number of embeddings of P1 into P2, where P1 and P2 are the partially ordered sets in the picture above Theorem 2.3.2.

2.4 Large implies tall or wide

Let (X,) be a finite partially ordered set. For brevity, let us denote it by the letter P . In most of this section we will consider only one (but arbitrary) ordered set. The notions that we will explore are introduced by the following definitions.

2.4.1 Definition. A set A ⊆ X is called independent in P if we never have x y for two distinct elements x, y ∈ A.

An independent set is also referred to as an antichain.

The definition can be rephrased using the following terminology.

Let us say that two distinct elements x and y are incomparable if neither x y, nor y  x. So a set is independent if every two of its elements are incomparable.

Let α(P ) denote the maximum size of an independent set in P . In symbols, this can be written

α(P ) = max{|A|: A independent in P }.

2.4.2 Example. For the following ordered sets P1 and P2

P1 P₂

we have α(P1) = 3, α(P2) = 4.

2.4.3 Observation. The set of all minimal elements in P is ind-ependent.

2.4.4 Definition. A set A ⊆ X is called a chain in P if every two of its elements are comparable (in P ).

Equivalently, the elements of A form a linearly ordered subset of P . Let ω(P ) denote the maximum number of elements of a chain in P . For the ordered sets P1 and P2 above we have ω(P1) = 3, ω(P2) = 2.

It is easy to check that ω(Bn) = n + 1. Determining α(Bn) is con-siderably more complicated; we answer this question in Chapter 7.

The above examples indicate that the number α(P ) can be thou-ght of as a kind of abstract “width” of the ordered set P , while ω(P ) corresponds to its “height”.

The following theorem, with an innocent-looking proof, actually has quite powerful consequences.

2.4.5 Theorem. For every finite ordered set P = (X,) we have α(P )· ω(P ) ≥ |X|.

(The reader may first want to check that the examples above fulfill the conclusion of the theorem.)

Proof. We define sets X1, X2, . . . , Xtinductively: Let X1 be the set of all minimal elements of the ordered set P . In an inductive step, let X1, . . . , X
be already defined, and let X
^ = X\

i=1X_i denote the set of all elements belonging to none of the sets X1, . . . , X
. If X
^ is the empty set, then we put t =  and the construction is finished.

Otherwise, for X
^ = ∅, we let ^ stand for the ordering restricted to the set X
^, and we define X
₊₁as the set of all minimal elements in (X
^,^). The proof will be finished as soon as we verify the following claims:

(1) The sets X1, . . . , X_t form a partition of X.

(2) Each X_i is an independent set in P . (3) ω(P )≥ t.

Claims (1) and (2) follow immediately from the definition of the sets X1, X2, . . . , X_t and from Observation 2.4.3. Thus, it suffices to prove (3).

By backward induction, for k = t, t− 1, . . . , 2, 1, we find elements x_i∈ X such that the set {x1, x2, . . . , x_t} constitutes a chain. Let us choose x_t∈ Xt arbitrarily. Since x_t∈ X/ t−1, there necessarily exists xt−1 ∈ Xt−1 so that xt−1 ≺ xt. This argument is a basis of the whole proof: Having already constructed elements xt ∈ Xt, xt−1 ∈ X_t−1, . . . , x_k+1 ∈ Xk+1, then x_k+1 ∈ X/ k, and hence there exists x_k∈ Xk with x_k ≺ xk+1.

2.4 Large implies tall or wide 57 The set{x1, . . . , xt} thus constructed is a chain. Therefore, ω(P )

≥ t. (Actually we have ω(P ) = t; we do not need this and we leave

it as a simple exercise.) 2

Theorem 2.4.5 has a number of nice connections, as is illustrated by the following celebrated application:

2.4.6 Theorem (Erd˝os–Szekeres lemma). An arbitrary sequence (x1, . . . , x_n2+1) of real numbers contains a monotone subsequence of length n + 1.

Let us first define the notions in this theorem explicitly. A subse-quence of length m is determined by indices i1, . . . , im, i1 < i2<· · ·

< i_m, and it has the form (x_i1, x_i2, . . . , x_im). Such a sequence is monotone if we have either x_i1 ≤ xi₂ ≤ · · · ≤ xi_m, or x_i1 ≥ xi₂ ≥

· · · ≥ xi_m. For example, the sequence (3, 5, 6, 2, 8, 1, 4, 7) contains the monotone subsequence (3, 5, 6, 8) (with i1 = 1, i2 = 2, i3 = 3 and i4 = 5), or the monotone subsequence (6, 2, 1) (with i1 = 3, i2 = 4, i3= 6), as well as many other monotone subsequences.

Proof of Theorem 2.4.6. Let a sequence (x1, . . . , x_n2+1) of n²+ 1 real numbers be given. Let us put X ={1, 2, . . . , n²+ 1}, and let us define a relation  on X by

i j if and only if both i ≤ j and xi ≤ xj.

It is not difficult to verify that the relation is a (partial) order-ing of the set X. So we have α(X,) · ω(X, ) ≥ n²+ 1, and hence α(X,) > n or ω(X, ) > n. Now it is easily checked that a chain i1≺ i2 ≺ · · · ≺ im in the ordering corresponds to a nondecreasing subsequence x_i1 ≤ xi₂ ≤ · · · ≤ xi_m (note that i1 < i2 < · · · < im), while an independent set{i¹, i2, . . . , i_m} corresponds to a decreasing subsequence (if we choose the notation so that i1 < i2 <· · · < im, then we get x_i1 > x_i2 >· · · > xi_m, since for example, x_i1 ≤ xi₂ and

i1< i2 would mean i1≺ i2). 2

Exercises

1. (a) Let i, i = 1, . . . , k, be orderings on some set X. Prove that k

i=1 i is again an ordering on X (recall that i is a relation, and thus a subset of X× X).

(b)^∗Prove that every partial ordering on a set X can be expressed as the intersection of linear orderings of X.

2. Prove that ω(Bn) = n + 1.

3. Find a sequence of real numbers of length 17 that contains no mono-tone subsequence of length 5.

4. Prove the following strengthening of Theorem 2.4.6: Let k,  be nat-ural numbers. Then every sequence of real numbers of length k + 1 contains an nondecreasing subsequence of length k + 1 or a decreasing subsequence of length  + 1.

5. (a) Prove that Theorem 2.4.5, as well as the preceding exercise, are optimal in the following sense: For every k and  there exists a partially ordered P with n elements such that n = k, α(P ) = k, and ω(P ) = .

(b)^∗Given k, ≥ 1, construct a sequence of real numbers of length k

with no nondecreasing subsequence of length k + 1 and no decreasing subsequence of length  + 1.

6. (a) Let us consider two sequences a = (a1, . . . , an) and b = (b1, . . . , bn) of distinct real numbers. Show that indices i1, . . . , ik, 1≤ i1 <· · · <

ik ≤ n, always exist with k = n^1/4 such that the subsequences determined by them in both a and b are increasing or decreasing (all 4 combinations are allowed, e.g. “increasing in a, decreasing in b”,

“decreasing in a, decreasing in b”, etc.).

(b)^∗Show that the bound for k in (c) cannot be improved in general.

7. ^∗∗(Dilworth’s theorem) Let (X,) be a finite partially ordered set.

Show that X can be expressed as a (disjoint) union of at most α = α(X,) chains.

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