Combinatorics Through Guided Discovery, 2004a

36 2. Applications of Induction and Recursion in Combinatorics and Graph Theory
⇒ ·
Problem 81. Prove that R(m, n) exists by proving that if there are ( m+n−2
m−1 )
people in a room, then there are either at least m mutual acquaintances or
at least n mutual strangers. (h)
· Problem 82. Prove that R(m, n) ≤ R(m − 1, n)+R(m, n − 1). (h)
⇒ ·
Problem 83.
(a) What does the equation in Problem 82 tell us about R(4, 4)?
∗ (b)
Consider 17 people arranged in a circle such that each person is acquainted
with the first, second, fourth, and eighth person to the right
and the first, second, fourth, and eighth person to the left. can you
find a set of four mutual acquaintances? Can you find a set of four
mutual strangers? (h)
(c) What is R(4, 4)?
Problem 84. (Optional) Prove the inequality of Problem 81 by induction on
m + n.
Problem 85. Use Stirling’s approximation (Problem 46) to convert the upper
bound for R(n, n) that you get from Problem 81 to a multiple of a power of
an integer.
2.1.6 A bit of asymptotic combinatorics
Problem 83 gives us an upper bound on R(n, n). A very clever technique due to
Paul Erdös, called the “probabilistic method,” will give a lower bound. Since both
bounds are exponential in n, they show that R(n, n) grows exponentially as n gets
large. An analysis of what happens to a function of n as n gets large is usually called
an asymptotic analysis. The probabilistic method, at least in its simpler forms,
can be expressed in terms of averages, so one does not need to know the language
of probability in order to understand it. We will apply it to Ramsey numbers in
the next problem. Combined with the result of Problem 83, this problem will give
us that √ 2 n < R(n, n) < 2 2n−2 , so that we know that the Ramsey number R(n, n)
grows exponentially with n.