Davide Leonessi, CUNY
Strategy and determinacy in infinite Hex
The popular game of Hex can be extended to the infinite hexagonal lattice, defining a winning condition which formalises the idea of a chain of colored stones stretching towards infinity. The descriptive-set-theoretic complexity of the set of winning positions is unknown, although it is at most Σ^1_1, and it is conjectured to be Borel; this has implications on whether games of infinite Hex are determined from all initial positions as either first-player wins or draws.
I will show that, unlike the finite game, infinite Hex with an initially empty board is a draw. But is the game still a draw when starting from a non-empty board? This open question can be partially answered in the positive by assuming the existence of certain local strategies, and in the negative by giving the advantage of placing two stones at each turn to one of the players. This is joint work with Joel David Hamkins.