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Published online by Cambridge University Press: 06 July 2010
The reader is owed a few missing details such as (1) how to modify the Achievement proofs to obtain the Avoidance proofs, (2) the Chooser-Picker game, (3) the bestknown Pairing Strategy Draw in the nd hypercube Tic-Tac-Toe (part (b) in Open Problem 34.1).
Also we discuss a few new results: generalizations and extensions, such as what happens if we extend the board from the complete graph KN and the N × N lattice to a typical sub-board.
We discuss these generalizations, extensions, and missing details in the last four sections (Sections 46–49).
More exact solutions and more partial results
Extension: from the complete board to a typical sub-board. The book is basically about two results, Theorems 6.4 and 8.2, and their generalizations (discrepancy, biased, Picker-Chooser, Chooser-Picker, etc.). Here is another, perhaps the most interesting, way to generalize. In Theorem 6.4 (a) the board is KN, that is, a very special graph; what happens if we replace KN with a typical graph GN on N vertices?
Playing the usual (1:1) game on an arbitrary finite graph G, we can define the Clique Achievement (Avoidance) Number of G in the usual way, namely answering the question: “What is the largest clique Kq that Maker can build (that Forcer can force Avoider to build)?”
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