Erdős-Ko-Rado for random hypergraphs: asymptotics and stability
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Marcelo M. Gauy and Hiêp Hàn and Igor C. Oliveira
2017
Abstract
We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for
the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq
n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability
tending to 1 as $n\to\infty$, the largest intersecting subhypergraph of
$\mathcal{H}^k(n,p)$ has size $(1+o(1))p\frac kn N$, for any $p\gg \frac
nk\ln^2\!\left(\frac nk\right)D^{-1}$. This lower bound on $p$ is
asymptotically best possible for $k=\Theta(n)$. For this range of $k$ and $p$,
we are able to show stability as well.
A different behavior occurs when $k = o(n)$. In this case, the lower bound on
$p$ is almost optimal. Further, for the small interval $D^{-1}\ll p \leq
(n/k)^{1-\varepsilon}D^{-1}$, the largest intersecting subhypergraph of
$\mathcal{H}^k(n,p)$ has size $\Theta(\ln (pD)N D^{-1})$, 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
$\mathcal{H}^k(n,p)$, for essentially all values of $p$ and $k$.
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