On the stability of the Erdős-Ko-Rado theorem
Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as an intersecting (uniform) family, this gives us a random analogue of the Erdős-Ko-Rado theorem.
Journal of Combinatorial Theory. Series A
Bollobás, B., Narayanan, B., & Raigorodskii, A. (2016). On the stability of the Erdős-Ko-Rado theorem. Journal of Combinatorial Theory. Series A, 137, 64-78. https://doi.org/10.1016/j.jcta.2015.08.002