We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial. We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model. Underpinning both results is a general combinatorial theorem that extends the recent work of Kelley, Lovett, and Meka (STOC'24), itself a development of ideas from the breakthrough result of Kelley and Meka on three-term arithmetic progressions (FOCS'23).

翻译：我们将准随机群中无角集密度的最佳已知上界从逆多对数改进为准多项式。类似地，我们对三玩家额上数字模型中一大类置换函数的通信复杂度下界作出了同等重大的改进。支撑这两项结果的是一般组合定理，它扩展了Kelley、Lovett和Meka（STOC'24）的最新工作，而该工作本身又是对Kelley和Meka在三项算术级数问题上突破性成果（FOCS'23）的思想发展。