Computability and Logic

320 RAMSEY’S THEOREM
Figure 26-1. A party with no clique or anticlique of three.
We are going to prove a theorem that bears on these puzzles. Recall that by a
partition of a nonempty set we mean a family of nonempty subsets thereof, called the
classes of the partition, such that every element of the original set belongs to exactly
one of these classes. By a size-k set we mean a set with exactly k elements.
26.1 Theorem (Ramsey’s theorem). Let r, s, n be positive integers with n ≥ r. Then
there exists a positive integer m ≥ n such that for X ={0, 1,...,m − 1}, no matter how
the size-r subsets of X are partitioned into s classes, there will always be a size-n subset Y
of X such that all size-r subsets of Y belong to the same class.
A set Y all of whose size-r subsets belong to the same one of the s classes is called
a homogeneous set for the partition. Note that if the theorem holds as stated, then it
clearly holds for any other size-m set in place of {0, 1,...,m − 1}.
For instance, it holds for the set of partiers at a party where m persons are present.
In the puzzles, the size-2 subsets of the set of persons at the party were partitioned
into two classes, one consisting of the pairs of persons who like each other, the other,
of the pairs of persons who dislike each other. So in both problems r = s = 2. In the
first, where n = 3, we showed how to prove that m = 6 is large enough to guarantee
the existence of a homogeneous set of size n—a clique of three who like each other,
or an anticlique of three who dislike each other. We also showed that 6 is the least
number m that is large enough. In the second problem, where n = 4, we reported that
m = 18 is large enough, and that 18 is in fact the least value of m that is large enough.
In principle, since there are only finitely many size-r subsets of {0,...,m − 1},
and only finitely many ways to partition these finitely many subsets into s classes,
and since there are only finitely many size-n subsets, we could set a computer to
work searching through all partitions, and for each looking for a homogeneous set.
If some partition were found without a homogeneous set, the computer could go on
to do a similar check for {0,...,m}. Continuing in this way, in a finite amount of
time it would find the least m that is large enough to guarantee the existence of the
required homogeneous set.
In practice, the numbers of possibilities to be checked are so large that such a
procedure is hopelessly infeasible. We do not at present have sufficient theoretical
insight into the problem to be able to reduce the number of possibilities that would
have to be checked to the point where a computer could feasibly be employed in
surveying them in order to pinpoint the least m. And it is entirely conceivable that
because of the such physical limitations as those imposed by the speed of light, the
atomic character of matter, and the short amount of time before the universe becomes
unable to sustain life, we are never going to know exact what the value of the least m
is, even for some quite small values of r, s, and n.