The partial order width of a set 
is equal to the minimum number of chains
needed to cover 
. Equivalently, if a set 
of 
elements is partially
ordered, then 
contains a chain of size 
or an antichain of size 
. Letting 
be the cardinal number
of 
,
the partial
order width, and 
the partial order length, this last statement
says 
.
Dilworth's lemma is a generalization of the Erdős-Szekeres
theorem. Ramsey's theorem generalizes Dilworth's
lemma.
Dilworth's Lemma
See also
Antichain, Chain, Combinatorics, Erdős-Szekeres Theorem, Ramsey's TheoremExplore with Wolfram|Alpha
References
Dilworth, R. P. "A Decomposition Theorem for Partially Ordered Sets." Ann. Math. 51, 161-166, 1950.Skiena, S. "Dilworth's Lemma." §6.4.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 241-243, 1990.Referenced on Wolfram|Alpha
Dilworth's LemmaCite this as:
Weisstein, Eric W. "Dilworth's Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DilworthsLemma.html