Zero-One Laws for Existential First-Order Sentences of Bounded Quantifier Depth

For any fixed positive integer k , let α denote the smallest ∈ (0,1) such that random graph sequence { G ( n, n -α )} does not satisfy zero-one law for set ε of all existential first-order sentences are quantifier depth at most . This article finds upper and lower bounds on showing as → ∞, we have = - 2 t )) -1 some function ) Θ -2 ). We also establish precise value when 4.