Dilworth's theorem 
[Nov. 25th, 200810:21 pm]
Yarrow

A set P is partially ordered if for every x, y, and z in P:
 Reflexivity: x is less than or equal to x;
 Antisymmetry: If x is less than or equal to y and y is less than or equal to x, then x equals y; and
 Transitivity: If x is less than or equal to y and y is less than or equal to z, then x is less than or equal to z.
If x is less than or equal to y or y is less than or equal to x, then we say that x and y are comparable. If it is neither the case that x is less than or equal to y or y is less than or equal to x, then we say x and y are incomparable.
A chain in P is a subset of P where every two members of the subset are comparable.
An antichain in P is a subset of P where no two members of the subset are comparable (unless they are equal).
The width of a finite partial order is the size of its largest antichain.
Dilworth's theorem says that the width of P is equal to the smallest n such that P can be written as the union of n chains.
I keep nibbling away at the proof of this one, but it's slippery.
Offering: Doors are solemn things, guarding entrances, Opening to the favored, Closing to thieves and weather. Doors are solemn things, but Cat doors are for cats. 

