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−ϵ
D
−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.