THE CRITICAL BIAS FOR THE HAMILTONICITY GAME IS (1 + o(1))n/ lnn

A Maker-Breaker game is a triple (H, a, b), where H = (V,E) is a hypergraph with vertex set V , called the board of the game, and edge set E, a family of subsets of V called winning sets. The parameters a and b are positive integers, related to the so-called game bias. The game is played between two players, called Maker and Breaker, who change turns occupying previously unclaimed elements of V ; Maker claims a elements in his turn, Breaker answers by claiming b elements. We assume that Breaker moves first. The game ends when all board elements have been claimed by either of the players. (In the very last move, if the board does not contain enough elements to claim for the player whose turn is now, that player claims all remaining elements of the board.) Maker wins if and only if he has occupied one of the winning sets e ∈ E by the end of the game. Breaker wins otherwise, i.e., if he manages to occupy at least one element of (“to break into”) every winning set by the end of the game. The most basic case is when a = b = 1, which is the so-called unbiased game. Here we will be concerned with 1 : b games. It is quite easy to see that Maker-Breaker games are bias monotone. This is to say that if the game (H, 1, b) is Maker’s win, then (H, 1, b′) is Maker’s win as well for every integer b′ < b. This allows us to define the critical bias of the game H, which is the maximum possible value of the bias b for which Maker still wins the 1 : b game played on H (if the 1:1 game is Breaker’s win, we say that the critical bias in this case is zero). We refer the reader to a recent monograph [2] of Beck for extensive background on positional games in general and on Maker-Breaker games in particular. The subject of this paper is the Hamiltonicity game played on the edge set of the complete graph Kn. In this game, players take turns in claiming unoccupied edges of Kn. Maker’s aim is to construct a Hamilton cycle, and thus the family of winning sets coincides with the family of (the edge sets of) graphs on n vertices containing a Hamilton cycle. The research on biased Hamiltonicity games has a long and illustrious history. Already in the very first paper about biased Maker-Breaker games back in 1978, Chvátal and Erdős [5] treated the unbiased Hamiltonicity game and showed that Maker wins it for every sufficiently large n. (Chvátal and Erdős showed in fact that Maker wins within 2n rounds. Later the minimum number of