Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability

We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph $\mathcal{H}$k(n, p). For 2⩽k(n) ⩽ n/2, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of $\mathcal{H}$ has size $$(1+o(1))p\ffrac kn N$$ for any $$p\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$ This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well.A different behaviour occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D−1 ≪ p ⩽ (n/k)1−ϵD−1, the largest intersecting subhypergraph of $\mathcal{H}$k(n, p) has size Θ(ln(pD)ND−1), provided that $k \gg \sqrt{n \ln n}$.Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}$k, for essentially all values of p and k.