 About 694 results  books.google.com These similarities are no coincidence. In fact, the ErdésSzekeres theorem may be
proved as a corollary of Dilworth's lemma. Let S = {a1,...,amn+1} be a sequence
of mn + 1 real numbers. Define a partial order _< on S by setting a, _< a] if a, ... 

 books.google.com But see also Dilworth's Lemma 3.2 of the paper before us. Thus Dilworth's insight
that relatively complemented lattices have permuting congruence relations
played a key role in establishing the decomposition into simple lattices. In
proving ... 

 books.google.com Dilworth's Lemma The WIDTH of a set P is equal to the minimum number of
CHAINS needed to COVER P. Equivalently, if a set P of ab'1 elements is
PARTIALLY ORDERED, then P contains a CHAIN of size a'1 or an ANTICHAIN of
size b'1: ... 

 books.google.com Derive the ErdösSzekeres theorem from Lemma 4.5. Hint: Given a sequence A =
(a1, ..., an) of n > r3 + 1 real numbers, define a partial order < on A by ai = a, if a s
aj and i < j, and apply Dilworth's lemma. 4.10. Let n° + 1 points be given in R*. 

 books.google.com We first recall Dilworth's Lemma, a duality theorem for posets: Lemma 9.10. (
Dilworth's Lemma) The width of a poset P is equal to the minimum number of
chains needed to cover P. (A family of nonempty subsets of a set Q is said to
cover Q if ... 

 books.google.com Together with Lovasz's theorem that the complement of a perfect graph is perfect;
36 this yields an indirect proof of Dilworth's theorem. 36Lovasz (1983). 3.2.5.
Width and order dimension.37 (a) Prove Dilworth's lemma. Let 3.2 Chains and ... 

 books.google.com As shown below, this covering can be proved using a geometric version of a
known combinatorial result (Dilworth's lemma, or ErdosSzekeres theorem).
Unfortunately, no equivalent combinatorial result is available in higher dimension
, and it ... 

 books.google.com ... then A ⊂ B and B ⊂ C, whence A ⊂ C, that is, A ≼ C. 7.3. The proof of
Dilworth's lemma is analogous to that of Theorem 7.2. With each element of the
given partially ordered set X, associate the length f(x) of a longest chain
beginning with x. 

 books.google.com The following is a direct corollary of Theorem 9.7.3 and Lemma 13.5.1. ...
Dilworth proved the following wellknown minmax result relating chains to
antichains: Theorem 13.5.8 (Dilworth's theorem) [260] Let P = (X,≺) be a partial
order. 

 books.google.com Dilworth's. Theorem. In this section we deal with decomposition theorems for
directed graphs and partially ordered sets. ... number of G.9 Obviously, the
complement of an independent set is a vertex cover; this implies the following
lemma. 

 