A new Bound for the Maker-Breaker Triangle Game

Abstract: The triangle game introduced by Chv\'{a}tal and Erd\H{o}s (1978) is one of the most famous combinatorial games. For $n,q\in\mathbb{N}$, the $(n,q)$-triangle game is played by two players, called Maker and Breaker, on the complete graph $K_n$. Alternately Maker claims one edge and thereafter Breaker claims $q$ edges of the graph. Maker wins the game if he can claim all three edges of a triangle, otherwise Breaker wins. Chv\'{a}tal and Erd\H{o}s (1978) proved that for $q<\sqrt{2n+2}-5/2\approx 1.414\sqrt{n}$ Maker has a winning strategy, and for $q\geq 2\sqrt{n}$ Breaker has a winning strategy. Since then, the problem of finding the exact leading constant for the threshold bias of the triangle game has been one of the famous open problems in combinatorial game theory. In fact, the constant is not known for any graph with a cycle and we do not even know if such a constant exists. Balogh and Samotij (2011) slightly improved the Chv\'{a}tal-Erd\H{o}s constant for Breaker's winning strategy from $2$ to $1.935$ with a randomized approach. Since then no progress was made. In this work, we present a new deterministic strategy for Breaker's win whenever $n$ is sufficiently large and $q\geq\sqrt{(8/3+o(1))n}\approx 1.633\sqrt{n}$, significantly reducing the gap towards the lower bound. In previous strategies Breaker chooses his edges such that one node is part of the last edge chosen by Maker, whereas the remaining node is chosen more or less arbitrarily. In contrast, we introduce a suitable potential function on the set of nodes. This allows Breaker to pick edges that connect the most `dangerous' nodes. The total potential of the game may still increase, even for several turns, but finally Breaker's strategy prevents the total potential of the game from exceeding a critical level and leads to Breaker's win.