Lecture 27 – Ramsey Numbers & Markov Chains

You may want to download the lecture slides that were used for these videos (PDF).

1. Complete Chaos

In the discrete world, complete chaos is impossible. Clear patterns exhibiting complete symmetry must emerge. (1:15)

2. Ramsey Numbers – Evidence of Uniformity

For positive integers m and n, the Ramsey number R(m, n) is the least positive integer t so that if G is any graph on t vertices, then either G contains a clique of size m or G contains an independent set of size n. (2:04)

3. Ramsey Numbers (1)

Examples: R(m, n) = R(n, m), R(m, 1) = 1 for all m, and R(m, 2) = m for all m. (8:36)

4. Ramsey Numbers (2)

Theorem: The Ramsey number R(m, n) exists and satisfies the inequality R(m, n) ≤ C(m + n – 2, m – 1). (5:23)

5. Small Ramsey Numbers

R(3, n) is known exactly for n ≤ 9. On the other hand, 40 ≤ R(3, 10) ≤ 42. For large n, it is now known that R(3, n) = Θ( n2/log n).

6. Suppose n = 12

We have that 2n/2 ≤ R(n,n) ≤  22n. If n = 12, what range of values can our Ramsey number have? (1:39)

7. Proving the Upper & Lower Bounds

We prove that 2n/2 ≤ R(n, n) ≤ 22n. The upper bound is easy to show, but the lower bound is more complicated. (8:43)

8. History of the Lower Bound

This video gives us some of the history history behind the development of the lower bound. (3:18)

9. A Challenge Worth a PhD

We have seen that 2n/2 ≤ R(n, n) ≤ 22n. Challenge: Move the constant in the exponent in either bound in the inequality for R(n, n). You will certainly have a marvelous PhD thesis as a result. (1:05)

10. Markov Chains – Examples and Questions

We illustrate an enclosure with 6 rooms. Suppose we start in the room in Room 1. We then move from room to room by the following rule: Once an hour, we choose at random one of the doors in the current room and exit to the adjoining room. What is the expected waiting time before we first reach Room 6? (7:28)

11. Absorbing Markov Chains

Suppose we start Room 1, but Room 6 has a tiger which will end our journey, what is our expected time of survival? (4:00)