Chapter III - Simple applications

Summary

The results formulated in the previous chapter (Chapter II) will be proved in Chapters V–IX, that is, we will need 5 chapters, more than 250 pages! Chapter III lays an intermediate role: it is a preparation for the main task, and also it answers some of the questions raised in Section 4. For example, in Section 15 we discuss an interesting result related to Kaplansky's n-in-a-line game.

The main goal of Chapter III is to demonstrate the amazing flexibility of the potential technique on a wide range of simple applications.

Easy building via Theorem 1.2

Some of the statements formulated in Chapter II have easy proofs. So far we proved two potential criterions, both simple: (1) the Weak Win criterion Theorem 1.2, and (2) the Strong Draw criterion Theorem 1.4 (“Erdős-Selfridge”). In a few lucky cases a direct reference to Theorem 1.2 supplies the optimal result.

Weak Win in the Van der Waerden Game. A particularly simple example is the upper bound in Theorem 8.1 (“arithmetic progression game”). We recall the (N, n) Van der Waerden Game: the board is [N] = {1, 2, … N} and the winning sets are the n-term A.P.s (“arithmetic progressions”) in [N].