In 1972, Branko Gr\"unbaum conjectured that any arrangement of $n>2$ pairwise crossing pseudocircles in the plane can have at most $2n-2$ digons (regions enclosed by exactly two pseudoarcs), with the bound being tight. While this conjecture has been confirmed for cylindrical arrangements of pseudocircles and more recently for geometric circles, we extend these results to any simple arrangement of pairwise intersecting pseudocircles. Using techniques from the above-mentioned special cases, we provide a complete proof of Gr\"unbaum's conjecture that has stood open for over five decades.

翻译：1972年，Branko Grünbaum提出猜想：平面上任意n>2个成对交叉的伪圆排列最多能形成2n-2个二边形（由恰好两条伪圆弧围成的区域），且该上界是紧的。虽然该猜想已在伪圆的柱面排列情形及近年来的几何圆情形中得到证实，但我们将这些结果推广至任意简单排列的成对相交伪圆。通过运用上述特殊情形中的技术方法，我们为这个悬置五十余年的Grünbaum猜想提供了完整证明。