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:
“If I could only live at the pitch that is near madness
When everything is as it was in my childhood
Violent, vivid, and of infinite possibility:
That the sun and the moon broke over my head.”
—Richard Eberhart (b. 1904)
“Remember that you were a slave in the land of Egypt, and the LORD your God brought you out from there with a mighty hand and an outstretched arm; therefore the LORD your God commanded you to keep the sabbath day.”
—Bible: Hebrew, Deuteronomy 5:15.
See Exodus 22:8 for a different version of this fourth commandment.
“Begin with loss and see
how the world contradicts you,
how the horizon implies that beyond it
the water is not empty
but full of ships
all docking at another island.”
—Lynn Emanuel (b. 1949)
“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)