The triangle game introduced by Chvátal and Erdős (1978) is one of the old and famous combinatorial games. For n, q ∈ 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. Chvátal and Erdős (1978) proved that for q < sqrt(n/2), Maker has a winning strategy, while for q > 2 sqrt(n), Breaker wins. So, the threshold bias must be in the interval [sqrt(1/2)sqrt(n) , 2 sqrt(n)].
Since then, the problem of finding the exact constant (and an associated Breaker strategy) for the threshold bias of the triangle game has been one of the interesting 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 Chvátal-Erdős constant for Breaker’s winning strategy from 2 to 1.935 with a randomized approach. Thereafter, no progress was made. In this work, we present a new deterministic strategy for Breaker leading to his win if q > sqrt(8/3) sqrt(n), for sufficiently large n. This almost matches the Chvátal-Erdős bound of sqrt(1/2)sqrt(n) for Maker's win (Glazik, Srivastav, Europ. J. Comb. 2022).
In contrast to previous (greedy) strategies we introduce a suitable non-linear potential function on the set of nodes. By keeping the potential small, Breaker picks edges that neutralize the most ‘dangerous’ nodes with incident Maker edges blocking Maker triangles. A characteristic property of the dynamics of the game is that the total potential is not monotone decreasing. In fact, the total potential of the game may increase, even for several turns, but finally Breaker’s strategy prevents the total potential of the game from exceeding a critical level, which results in Breaker’s win. We further survey recent results for cycles of length k, and a general potential function theorem (Sowa, Srivastav 2023).

This is joint work with Christian Glazik, Christian Schielke and Mathias Sowa, Kiel University.