The pairing strategies of the 9-in-a-row game - Győrffy, Lajos (Ars mathematica contemporanea, 2019)
One of the most useful strategies for proving Breaker's win in Maker-Breaker Positional Games is to find a pairing strategy. In some cases there are no pairing strategies at all, in some cases there are unique or almost unique strategies. For the ?$k$?-in-a-row game, the case ?$k = 9$? is the smallest (sharp) for which there exists a Breaker winning pairing (paving) strategy. One pairing strategy for this game was given by Hales and Jewett. In this paper we show that there are other winning pairings for the 9-in-a-row game, all have a very symmetric torus structure. While describing these symmetries we prove that there are only a finite number of non-isomorphic pairings for the game (around 200 thousand), which can be also listed up by a computer program. In addition, we prove that there are no "irregular", non-symmetric pairings. At the end of the paper we also show a pairing strategy for a variant of the 3-dimensional ?$k$?-in-a-row game.
Keywords: igra k-v-vrsti; k-in-a-row game; pairing strategies; positional games; pozicijska igra; simetrije; strategije prirejanja; symmetries