Consider a host hypergraph $G$ which contains a spanning structure due to minimum degree considerations. We collect three results proving that if the edges of $G$ are sampled at the appropriate rate then the spanning structure still appears with high probability in the sampled hypergraph. We prove such results for perfect matchings in hypergraphs above Dirac thresholds, for $K_r$-factors in graphs satisfying the Hajnal--Szemer\'edi minimum degree condition, and for bounded-degree spanning trees. In each case our proof is based on constructing a spread measure and then applying recent results on the (fractional) Kahn--Kalai conjecture connecting the existence of such measures with probabilistic thresholds. For our second result we give a shorter and more general proof of a recent theorem of Allen, B\"ottcher, Corsten, Davies, Jenssen, Morris, Roberts, and Skokan which handles the $r=3$ case with different techniques. In particular, we answer a question of theirs with regards to the number of $K_r$-factors in graphs satisfying the Hajnal--Szemer\'edi minimum degree condition.

翻译：考虑一个由最小度条件保证包含某类跨越结构的主超图$G$。我们收集了三项结果，证明若以适当速率对$G$的边进行采样，则采样后的超图以高概率仍包含该跨越结构。具体结果涵盖：达到迪拉克阈值以上的超图中的完美匹配，满足Hajnal–Szemerédi最小度条件的图中的$K_r$-因子，以及有界度生成树。每个证明均基于构造扩散测度，并应用近期关于Kahn–Kalai猜想（分数版本）将此类测度存在性与概率阈值相联系的结果。针对第二项结果，我们给出了Allen、Böttcher、Corsten、Davies、Jenssen、Morris、Roberts与Skokan近期定理的更简短且更普适的证明——该定理采用不同技术处理了$r=3$情形。特别地，我们回应了他们对满足Hajnal–Szemerédi最小度条件的图中$K_r$-因子数量提出的一个疑问。