A robust Corrádi–Hajnal theorem

For a graph (Formula presented.) and (Formula presented.), we denote by (Formula presented.) the random sparsification of (Formula presented.) obtained by keeping each edge of (Formula presented.) independently, with probability (Formula presented.). We show that there exists a (Formula presented.) such that if (Formula presented.) and (Formula presented.) is an (Formula presented.) -vertex graph with (Formula presented.) and (Formula presented.), then with high probability (Formula presented.) contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of (Formula presented.), are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corrádi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to (Formula presented.) in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in (Formula presented.) (corresponding to (Formula presented.) in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least (Formula presented.) which gets close to the truth.