Infinite Version Implies The Finite
It is possible to deduce the finite Ramsey theorem from the infinite version by a proof by contradiction. Suppose the finite Ramsey theorem is false. Then there exist integers such that for every integer, there exists a -colouring of without a monochromatic set of size . Let denote the -colourings of without a monochromatic set of size .
For any k, the restriction of a colouring in to (by ignoring the colour of all sets containing ) is a colouring in . Define to be the colourings in which are restrictions of colourings in . Since is not empty, neither is .
Similarly, the restriction of any colouring in is in, allowing one to define as the set of all such restrictions, a non-empty set. Continuing so, define for all integers .
Now, for any integer, and each set is non-empty. Furthermore, is finite as . It follows that the intersection of all of these sets is non-empty, and let . Then every colouring in is the restriction of a colouring in . Therefore, by unrestricting a colouring in to a colouring in, and continuing doing so, one constructs a colouring of without any monochromatic set of size . This contradicts the infinite Ramsey theorem.
If a suitable topological viewpoint is taken, this argument becomes a standard compactness argument showing that the infinite version of the theorem implies the finite version.
Read more about this topic: Ramsey's Theorem
Famous quotes containing the words infinite, version, implies and/or finite:
“This moment exhibits infinite space, but there is a space also wherein all moments are infinitely exhibited, and the everlasting duration of infinite space is another region and room of joys.”
—Thomas Traherne (1636–1674)
“If the only new thing we have to offer is an improved version of the past, then today can only be inferior to yesterday. Hypnotised by images of the past, we risk losing all capacity for creative change.”
—Robert Hewison (b. 1943)
“Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.”
—David Hume (1711–1776)
“The finite is annihilated in the presence of the infinite, and becomes a pure nothing. So our spirit before God, so our justice before divine justice.”
—Blaise Pascal (1623–1662)