The missing Strong Draw parts of Theorems 8.2, 12.6, and 40.2 will be discussed here; we prove them in the reverse order. These are the most difficult proofs in the book. They demand a solid understanding of Chapter VIII. The main technical challenge is the lack of Almost Disjointness.

Chapters I–VI were about Building and Chapters VII–VIII were about Blocking. We separated these two tasks because undertaking them at the same time – ordinary win! – was hopelessly complicated. Now we have a fairly good understanding of Building (under the name of Weak Win), and have a fairly good understanding of Blocking (under the name of Strong Draw). We return to an old question one more time: “Even if ordinary win is hopeless, is there any other way to combine the two different techniques in a single strategy?” The answer is “yes,” and some interesting examples will be discussed in Section 45. One of them is the proof of Theorem 12.7: “second player's moral-victory.”

Winning planes: exact solution

The objective of this section is to prove the missing Strong Draw part of Theorems 12.6 and 40.2. The winning sets in these theorems are “planes”; two “planes” may be disjoint, or intersect in a point, or intersect in a “line.” The third case – “line-intersection” – is a novelty which cannot happen in Almost Disjoint hypergraphs; “line-intersection” requires extra considerations.