Combinatorics Through Guided Discovery, 2004a

26 1. What is Combinatorics?
· Problem 63. There is a generalized pigeonhole principle which says that
if we partition a set with more than kn elements into n blocks, then at least
one block has at least k +1elements. Prove the generalized pigeonhole
principle. (h)
Problem 64. All the powers of five end in a five, and all the powers of two
are even. Show that for for some integer n, if you take the first n powers of
a prime other than two or five, one must have “01” as the last two digits. (h)
⇒ ·
Problem 65. Show that in a set of six people, there is a set of at least three
people who all know each other, or a set of at least three people none of
whom know each other. (We assume that if person 1 knows person 2, then
person 2 knows person 1.) (h)
· Problem 66. Draw five circles labeled Al, Sue, Don, Pam, and Jo. Find a
way to draw red and green lines between people so that every pair of people
is joined by a line and there is neither a triangle consisting entirely of red
lines or a triangle consisting of green lines. What does Problem 65 tell you
about the possibility of doing this with six people’s names? What does this
problem say about the conclusion of Problem 65 holding when there are
five people in our set rather than six?
1.3.4 Ramsey Numbers
Problems 65–66 together show that six is the smallest number R with the property
that if we have R people in a room, then there is either a set of (at least) three
mutual acquaintances or a set of (at least) three mutual strangers. Another way to
say the same thing is to say that six is the smallest number so that no matter how
we connect 6 points in the plane (no three on a line) with red and green lines, we
can find either a red triangle or a green triangle. There is a name for this property.
The Ramsey Number R(m, n) is the smallest number R so that if we have R people
in a room, then there is a set of at least m mutual acquaintances or at least n mutual
strangers. There is also a geometric description of Ramsey Numbers; it uses the
idea of a complete graph on R vertices. A complete graph on R vertices consists of
R points in the plane together with line segments (or curves) connecting each two
of the R vertices.1 The points are called vertices and the line segments are called
edges. InFigure 1.3.2 we show three different ways to draw a complete graph on
four vertices. We use K n to stand for a complete graph on n vertices.
1As you may have guessed, a complete graph is a special case of something called a graph. The word
graph will be defined in Subsection 2.3.1.