The union-closed sets conjecture, attributed to P\'eter Frankl from 1979, states that for any non-empty finite union-closed family of finite sets not consisting of only the empty set, there is an element that is in at least half of the sets in the family. We prove Frankl's conjecture for families distributed according to any one of infinitely many distributions. As a corollary, in the intersection-closed reformulation of Frankl's conjecture, we obtain that it is true for families distributed according to any one of infinitely many Maxwell--Boltzmann distributions with inverse temperatures bounded below by a positive universal constant. Frankl's original conjecture corresponds to zero inverse temperature.

翻译：并封闭集猜想，由Péter Frankl于1979年提出，断言：对于任意非空且仅由有限集构成的并封闭集族（且该族不只有空集），存在一个元素至少出现在族中一半的集合里。我们证明了Frankl猜想对于按无穷多种分布中任意一种分布的集族成立。作为推论，在Frankl猜想的交封闭重构形式中，我们得到其对于按无穷多种麦克斯韦-玻尔兹曼分布中任意一种分布的集族成立，这些分布的反向温度被一个正的普适常数所下界限制。Frankl原猜想对应的是零反向温度的情况。