Zero-One Law for random uniform hypergraphs

In this work limit probabilities of first-order properties of the random $s$-uniform hypergraph in the binomial model $G^s(n,p)$ are studied. We give a complete discription of all positive $\alpha$ such that $G^s(n,n^{-\alpha })$ obeys Zero-One Law. Moreover, for any rational $\rho\geq 1/(s-1)$weprovetheexistenceofastrictlybalanced$s$-uniform hypergraph with the density $\rho$.