4 - Looking Back

Summary

In this chapter, we look back at some of the problems and their solutions, in both this volume and in Hungarian Problem Book III. We look at various extensions and generalizations as well as related results. While this material is not needed for solving the problems at hand, they may provide valuable insight in solving other problems.

Discussion on Combinatorics

Most of our discussion in this section is on graph theory. We begin by looking back at the following problem which has become a classic.

Problem 1947.2

Prove that in any group of six people, either there are three people who know one another or three people who do not know one another. Assume that “knowing” is a symmetric relation.

This problem can be rephrased in graph theoretic terms in at least two ways other than that used in its solution. In the first version, which has a symmetric form, we have a graph on six vertices. We wish to conclude that either the graph or its complement contains a complete subgraph on 3 vertices.

This can be generalized as follows. What is the minimum number m(k) such that for any graph with m(k) vertices, either the graph or its complement contains a complete subgraph on k vertices?

Clearly, m(1) = 1 and m(2) = 2. The result of Problem 1947.2 shows that m(3) ≤ 6. In fact, m(3) ≤ 6 since neither a pentagon nor its complement contains a triangle.