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

  • MARCELO M. GAUY (a1), HIÊP HÀN (a2) and IGOR C. OLIVEIRA (a3)

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 −1p ⩽ (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.

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  • ISSN: 0963-5483
  • EISSN: 1469-2163
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