General independence sets in random strongly sparse hypergraphs

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 $p=c\big/\binom{n-1}{k-1}$ for a positive constant $0<c\le1/(k-1)$, there exists a constant $\gamma(k,j,c)>0$ such that the $j$-independence number $\alpha_j(H(n,k,p))$ obeys the law of large numbers
$$ \frac{\alpha_j(H(n,k,p))}{n}\xrightarrow{\mathbf P\,}\gamma(k,j,c)\qquad\text{as}\quad n\to+\infty. $$

Moreover, we explicitly present $\gamma(k,j,c)$ as a function of a solution of some transcendental equation.