Computability and Logic

26.2. K ÖNIG’S LEMMA 323
called the levels of the tree; and (iii) a two-place relation R subject to the following
conditions:
(1) Rab never holds for b in T 0 .
(2) For b in T n+1 , Rab holds for exactly one a, and that a is in T n .
When Rab holds, we say a is immediately below b, and b is immediately above a.
Figure 26-2 is a picture of a finite tree with ten nodes and four levels. Line segments
connect nodes immediately below and above each other.
Figure 26-2. A finite tree.
A branch through a tree is a sequence of nodes b 0 , b 1 , b 2 , ...with each b n immediately
below b n+1 . Obviously, an infinite tree none of whose levels is infinite must
have infinitely many nonempty levels. The following is not so obvious.
26.3 Lemma (König’s lemma). An infinite tree none of whose levels is infinite must
have an infinite branch.
Postponing the proof of this result, let us see how it can be used as a bridge between
the finite and the infinite.
Proof of Theorem 26.1: Suppose that Theorem 26.1 fails. Then for some positive
integers r, s, n, with n ≥ r, for every m ≥ n there exists a partition
f :[{0, 1,...,m − 1}] r →{1,...,s}
having no size-n homogeneous set Y . Let T be the set of all such partitions without
size-n homogeneous sets for all m, and let T k be the subset of T consisting of those
f with m = n + k. Let Rfg hold if and only if for some k
f :[{0, 1,...,n + k − 1}] r →{1,...,s}
g :[{0, 1,...,n + k}] r →{1,...,s}
and g extends f, in the sense that g assigns the same value as does f to any argument
in the domain of f . It is easily seen that for any g in T k+1 there is exactly one f in T k
that g extends, so what we have defined is a tree.
There are only finitely many functions from a given finite set to a given finite set,
so there are only finitely many nodes f in any level T k . But our initial supposition was
that for every m = n + k there exists a partition f in T k , so the level T k is nonempty for
all k, and the tree is infinite. König’s lemma then tells us there will be an infinite branch
f 0 , f 1 , f 2 ,...,which is to say, an infinite sequence of partitions, each extending the
one before, and none having a size-n homogenous set. We can then define a partition
F :[ω] r →{1,...,s}