We analyze the asymptotic behavior of the j-independence number of a random k-uniform hypergraph H(n, k, p) in the binomial model. We prove that in the strongly sparse case, i.e., where \(p = c/\left( \begin{gathered} n - 1 \hfill \\ k - 1 \hfill \\ \end{gathered} \right)\) for a positive constant 0 < c ≤ 1/(k − 1), there exists a constant γ(k, j, c) > 0 such that the j-independence number α _j (H(n, k, p)) obeys the law of large numbers \(\frac{{{\alpha _j}\left( {H\left( {n,k,p} \right)} \right)}}{n}\xrightarrow{P}\gamma \left( {k,j,c} \right)asn \to + \infty \) Moreover, we explicitly present γ(k, j, c) as a function of a solution of some transcendental equation.

中文翻译：

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随机强稀疏超图中的一般独立集

我们分析了二项式模型中随机k一致超图H（n，k，p）的j独立数的渐近行为。我们证明在极稀疏的情况下，即\（p = c / \ left（\ begin {gathered} n-1 \ hfill \\ k-1 \ hfill \\ \ end {gathered} \ right）\）为正的常数，0 < c ^ ≤1 /（ķ - 1），存在一个常数γ（ķ，Ĵ，ç）> 0，使得Ĵ -independence数α _Ĵ（ħ（ñ，ķ，p））服从大数定律 \（\ frac {{{{\ alpha _j} \ left（{H \ left（{n，k，p} \ right）} \ right）}} {n} \ xrightarrow {P} \ gamma \ left（{k ，j，c} \ right）asn \ to + \ infty \）此外，我们明确地提出γ（k，j，c）作为某些先验方程解的函数。