Computability and Logic

26
Ramsey’s Theorem
Ramsey’s theorem is a combinatorial result about finite sets with a proof that has interesting
logical features. To prove this result about finite sets, we are first going to prove,
in section 26.1, an analogous result about infinite sets, and are then going to derive, in
section 26.2, the finite result from the infinite result. The derivation will be an application
of the compactness theorem. Nothing in the proof of Ramsey’s theorem to be presented
requires familiarity with logic beyond the statement of the compactness theorem, but at
the end of the chapter we indicate how Ramsey theory provides an example of a sentence
undecidable in P that is more natural mathematically than any we have encountered
so far.
26.1 Ramsey’s Theorem: Finitary and Infinitary
There is an old puzzle about a party attended by six persons, at which any two of the
six either like each other or dislike each other: the problem is to show that at the party
there are three persons, any two of whom like each other, or there are three persons,
any two of whom dislike each other.
The solution: Let a be one of the six. Since there are five others, either there will be
(at least) three others that a likes or there will be three others that a dislikes. Suppose
a likes them. (The argument is similar if a dislikes them.) Call the three b, c, d. Then
if (case 1) b likes c or b likes d or c likes d, then a, b, and c, ora, b, and d, ora, c,
and d, respectively, are three persons any two of whom like each other; but if (case 2)
b dislikes c, b dislikes d, and c dislikes d, then b, c, and d are three persons, any two
of whom dislike each other. And either case 1 or case 2 must hold.
The number six cannot in general be reduced; if only five persons, a, b, c, d, e are
present, then the situation illustrated in Figure 26-1 can arise. (A broken line means
‘likes’; a solid line, ‘dislikes’.) In this situation there are no three of a, b, c, d, e any
two of whom like each other (a ‘clique’) and no three, any two of whom dislike each
other (an ‘anticlique’).
A harder puzzle of the same type is to prove that at any party such as the previous
one at which eighteen persons are present, either there are four persons, any two of
whom like each other, or four persons, any two of whom dislike each other. (This
puzzle has been placed among the problems at the end of this chapter.) It is known
that the number eighteen cannot be reduced.
319