A Toolkit for Robust Thresholds

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.