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：我们将拟随机群中角形自由集密度的已知最佳上界从逆多对数改进至拟多项式。对于3玩家额写数字模型中一大类置换函数的通信复杂度，我们同样显著改进了已知最佳下界。这两项结果的共同基础是一个广义组合定理，该定理扩展了Kelley、Lovett和Meka（STOC'24）的最新工作——其本身是对Kelley与Meka在三项算术级数问题上突破性成果（FOCS'23）的思想发展。