The main objective of Chapter VIII is to develop a more sophisticated version of the BigGame–SmallGame Decomposition technique (introduced in Sections 35–36).

We prove the second Ugly Theorem; We formulate and prove the third Ugly Theorem. Both are about Almost Disjoint hypergraphs. In Section 42 we extend the decomposition technique from Almost Disjoint to more general hypergraphs. We call it the RELARIN technique. These tools will be heavily used again in Chapter IX to complete the proof of Theorem 8.2.

Proof of the second Ugly Theorem

The Neighborhood Conjecture (Open Problem 9.1) is a central issue of the book. The first result toward Open Problem 9.1 was Theorem 34.1, or as we called it: the first Ugly Theorem (see Section 36 for the proof). The second Ugly Theorem (Theorem 37.5) is more powerful. It gives the best-known Strong Draw result for the nd hypercube Tic-Tac-Toe (Theorem 12.5 (a)), and it is also necessary for the solution of the Lattice Games (Theorem 8.2).

Proof of Theorem 37.5. We assume that the reader is familiar with the proof of Theorem 34.1. In the proof of Theorem 34.1 Breaker used the Power-of-Two Scoring System in the Big Game to prevent the appearance of the “Forbidden Configurations” in the small game, and this way he could ensure the “simplicity” of the small game. The small game was so simple that Breaker could block every “emergency set” by a trivial Pairing Strategy.