Given a k-uniform hypergraph ℋ and sufficiently large m ≫ m₀(ℋ), we show that an m-element set I ⊆ V(ℋ), chosen uniformly at random, with probability 1 − e−^ω(m) is either not independent or is contained in an almost-independent set in ℋ which, crucially, can be constructed from carefully chosen o(m) vertices of I. As a corollary, this implies that if the largest almost-independent set in ℋ is of size o(v(ℋ)) then I itself is an independent set with probability e^−ω(m). More generally, I is very likely to inherit structural properties of almost-independent sets in ℋ.

The value m₀(ℋ) coincides with that for which Janson’s inequality gives that I is independent with probability at most \({e^{- \Theta ({m_0})}}\). On the one hand, our result is a significant strengthening of Janson’s inequality in the range m ≫ m₀. On the other hand, it can be seen as a probabilistic variant of hypergraph container theorems, developed by Balogh, Morris and Samotij and, independently, by Saxton and Thomason. While being strictly weaker than the original container theorems in the sense that it does not apply to all independent sets of size m, it is nonetheless sufficient for many applications and admits a short proof using probabilistic ideas.

给定一个k -均匀超图 ℋ 和足够大的m ≫ m ₀ (ℋ)，我们证明 m 元素集I ⊆ V (ℋ)，均匀随机选择，概率为 1 − e− ^{ω ( m )}是不独立或者包含在 ℋ 中的一个几乎独立的集合中，至关重要的是，它可以从I的精心选择的o ( m ) 个顶点构造而成。作为推论，这意味着如果 ℋ 中最大的几乎独立集合的大小为o ( v (ℋ)) ，那么I本身就是概率为e ^{−ω ( m )}的独立集合。更一般地说，I很可能继承 ℋ 中几乎独立集合的结构属性。

值m ₀ (ℋ) 与 Janson 不等式给出的值一致，即I与概率至多\({e^{- \Theta ({m_0})}}\)无关。一方面，我们的结果是 Janson 不等式在m ≫ m ₀范围内的显着加强。另一方面，它可以被视为超图容器定理的概率变体，由 Balogh、Morris 和 Samotij 以及 Saxton 和 Thomason 独立开发。虽然它并不适用于所有大小为m的独立集合，但它比原始容器定理严格弱，但它对于许多应用来说仍然足够，并且允许使用概率思想进行简短证明。