Infinite Increasing Subsequences
If (, ≤) is wqo then every infinite sequence, … contains an infinite increasing subsequence ≤≤≤… (with <<<…). Such a subsequence is sometimes called perfect. This can be proved by a Ramsey argument: given some sequence, consider the set of indexes such that has no larger or equal to its right, i.e., with . If is infinite, then the -extracted subsequence contradicts the assumption that is wqo. So is finite, and any with larger than any index in can be used as the starting point of an infinite increasing subsequence.
The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering, leading to an equivalent notion.
Read more about this topic: Well-quasi-ordering
Famous quotes containing the words infinite and/or increasing:
“Mary McDonald, you giggled as you passed—
I wondered what the boy with hairy chest
Carved on the wall of his inexpensive spirit
Memorial to your infinite unrest.”
—Allen Tate (1899–1979)
“A point has been reached where the peoples of the Americas must take cognizance of growing ill-will, of marked trends toward aggression, of increasing armaments, of shortening tempers—a situation which has in it many of the elements that lead to the tragedy of general war.... Peace is threatened by those who seek selfish power.”
—Franklin D. Roosevelt (1882–1945)