Computability and Logic

326 RAMSEY’S THEOREM
which tells us that glorified Ramsey’s theorem is undecidable in P, requires a deeper
analysis of nonstandard models than that undertaken in the preceding chapter, and is
beyond the scope of a book such as this one.
Problems
26.1 Show that at a party attended by at least nine persons, any two of whom either
like each other or dislike each other, either there are four, any two of whom like
each other, or there are three, any two of whom dislike each other.
26.2 Show that at a party attended by at least eighteen persons, any two of whom
either like each other or dislike each other, either there are four, any two of
whom like each other, or there are four, any two of whom dislike each other.
26.3 A finite set of points in the plane, none lying on the line between any other two,
is said to be convex if no point lies in the interior of the triangle formed by any
three other points, as on the left in Figure 26-3. It is not hard to show that given
Figure 26-3. Convex and concave sets of points.
any set of five points in the plane, none lying on the line between any other
two, there is a convex subset of four points. The Erdös–Szekeres theorem states
that, more generally, for any number n > 4 there exists a number m such that
given a set of (at least) m points in the plane, none lying on the line between
any other two, there is a convex subset of (at least) n points. Show how this
theorem follows from Ramsey’s theorem.
26.4 Show that the general case of Ramsey’s theorem follows from the special case
with s = 2, by induction on s.
26.5 For r = s = 2 and n = 3, each node in the tree used in the proof of Theorem
26.1 in section 26.2 can be represented by a picture in the style of Figure 26-1.
How many such nodes will there be in the tree?
26.6 Prove König’s lemma directly, that is, without using the compactness theorem,
by considering the subtree T *ofT consisting of all nodes that have infinitely
many nodes above them (where above means either immediately above, or
immediately above something immediately above, or ...).