A combinatorial object is said to be quasirandom if it exhibits certain properties that are typically seen in a truly random object of the same kind. It is known that a permutation is quasirandom if and only if the pattern density of each of the twenty-four 4-point permutations is close to 1/24, which is its expected value in a random permutation. In other words, the set of all twenty-four 4-point permutations is quasirandom-forcing. Moreover, it is known that there exist sets of eight 4-point permutations that are also quasirandom-forcing. Breaking the barrier of linear dependency of perturbation gradients, we show that every quasirandom-forcing set of 4-point permutations must have cardinality at least five.

翻译：若一个组合对象展现出同类真随机对象中通常具备的某些性质，则称该对象是拟随机的。已知一个排列是拟随机的，当且仅当所有24种四点排列的图案密度均接近1/24——即其在随机排列中的期望值。换言之，全体24种四点排列的集合是拟随机强制的。此外，已知存在由八种四点排列构成的集合同样具有拟随机强制性。通过突破扰动梯度线性依赖性的壁垒，我们证明了任何具有拟随机强制性的四点排列集合的基数必须至少为五。