We prove the three candidate Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell from 2005 for correlations $\rho$ satisfying $-1/36<\rho<1/11$: the Plurality function is the most noise stable three candidate election method with small influences, when the corrupted votes have correlation $-1/36<\rho<1/11$ with the original votes. The previous best result of this type only achieved positive correlations at most $10^{-10^{10}}$. Our result follows by solving the three set Standard Simplex Conjecture of Isaksson-Mossel from 2011 for all correlations $-1/36<\rho<1/11$. The Gaussian Double Bubble Theorem corresponds to the case $\rho\to1^{-}$, so in some sense, our result is a generalization of the Gaussian Double Bubble Theorem. Our result is also notable since it is the first result for any $\rho<0$, which is the only relevant case for computational hardness of MAX-3-CUT. In fact, assuming the Unique Games Conjecture, we show that MAX-3-CUT is NP-hard to approximate within a multiplicative factor of $.9875$, which improves on the known (unconditional) NP-hardness of approximation within a factor of $1-(1/102)$, proven in 1997. As an additional corollary, we conclude that three candidate Borda Count is stablest for all $-1/36<\rho<1/11$.

翻译：我们证明了Khot-Kindler-Mossel-O'Donnell于2005年提出的三候选人简单多数制最稳定性猜想在相关系数$\rho$满足$-1/36<\rho<1/11$时成立：当被篡改选票与原选票的相关系数处于$-1/36<\rho<1/11$区间时，简单多数制函数是低影响力条件下噪声稳定性最优的三候选人选举方法。此前该问题的最佳结果仅能达到$10^{-10^{10}}$量级下的正相关系数。我们的成果通过求解Isaksson-Mossel于2011年提出的三集合标准单纯形猜想在相关系数$-1/36<\rho<1/11$范围内的全部情况而获得。高斯双泡定理对应$\rho\to1^{-}$的极端情形，因此本结果在某种意义上是对高斯双泡定理的推广。本成果的另一个重要价值在于首次涵盖了$\rho<0$的情形，而这正是MAX-3-CUT计算复杂性难题中唯一具有实际意义的场景。具体而言，在唯一游戏猜想成立的前提下，我们证明了MAX-3-CUT在乘法因子$.9875$内是NP难逼近的，这改进了1997年证实的已知（无条件）$1-(1/102)$因子NP难逼近结果。作为附加推论，我们推论波达计数法在所有$-1/36<\rho<1/11$条件下具有最稳定性。