Combinatorics Through Guided Discovery, 2004a

1.3. Some Applications of the Basic Principles 27
Figure 1.3.2: Three ways to draw a complete graph on four vertices
Our geometric description of R(3, 3) may be translated into the language of
graph theory (which is the subject that includes complete graphs) by saying R(3, 3)
is the smallest number R so that if we color the edges of a K R with two colors,
then we can find in our picture a K 3 all of whose edges have the same color. The
graph theory description of R(m, n) is that R(m, n) is the smallest number R so
that if we color the edges of a K R with red and green, then we can find in our
picture either a K m all of whose edges are red or a K n all of whose edges are green.
Because we could have said our colors in the opposite order, we may conclude that
R(m, n) =R(n, m). In particular R(n, n) is the smallest number R such that if we
color the edges of a K R with two colors, then our picture contains a K n all of whose
edges have the same color.
◦ Problem 67. Since R(3, 3) = 6, an uneducated guess might be that R(4, 4) =
8. Show that this is not the case. (h)
· Problem 68. Show that among ten people, there are either four mutual
acquaintances or three mutual strangers. What does this say about R(4, 3)? (h)
· Problem 69. Show that among an odd number of people there is at least one
person who is an acquaintance of an even number of people and therefore
also a stranger to an even number of people. (h)
· Problem 70. Find a way to color the edges of a K 8 with red and green so
that there is no red K 4 and no green K 3 . (h)
⇒ ·
Problem 71.
Find R(4, 3). (h)
As of this writing, relatively few Ramsey Numbers are known. R(3, n) is known
for n < 10, R(4, 4) = 18, and R(5, 4) = R(4, 5) = 25.