General Independence Sets in Random Strongly Sparse Hypergraphs

Abstract

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 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.