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:
“The unique eludes us; yet we remain faithful to the ideal of it; and in spite of sense and of our merely abstract thinking, it becomes for us the most real thing in the actual world, although for us it is the elusive goal of an infinite quest.”
—Josiah Royce (1855–1916)
“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)
“To insult a friend implies that you respect his masculinity enough to know he can take it without acting like a crybaby. The swapping of insults, like the fighting between brothers, becomes the seal of the male bonding.”
—Frank Pittman (20th century)
“God is a being of transcendent and unlimited perfections: his nature therefore is incomprehensible to finite spirits.”
—George Berkeley (1685–1753)