A Toolkit for Robust Thresholds

Consider a host (hyper)graph $G$ which contains a spanning structure due to minimum degree considerations. We collect three results proving that when 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 dense hypergraphs above Dirac thresholds, for $K_r$-factors above 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 an appropriate probabilistic threshold result. We note that our second result provides a shorter and more general version of a recent result of Allen, B\"ottcher, Corsten, Davies, Jenssen, Morris, Roberts, and Skokan which handles the case $r=3$ with different techniques. In particular, we answer a question of theirs with regards to the number of $K_r$-factors in a graph above the Hajnal--Szemer\'edi minimum degree condition.