The generic-and-Hilton–Milner obstruction conjecture for EKR in random hypergraphs

Let ${\cal H}$ be the random $k$-uniform hypergraph with edge-probability $\varphi$, let $\Delta$ denote its maximum degree, and let $\Lambda'(\Delta)$ be the parameter from Theorem~. Assume that

$$\Lambda'(\Delta)<o(1)\quad\text{a.s.}$$

and that ${\cal H}$ almost surely does not contain a Hilton–Milner family of size $\Delta$. Generic-and-Hilton–Milner obstruction conjecture. Under these assumptions, ${\cal H}$ almost surely satisfies the Erdős–Ko–Rado property.