On Failure of 0-1 Laws

Abstract

Let $\alpha \in (0,1)_{{\mathbb {R}}}$ be irrational and $G_n = G_{n,1/n^\alpha }$ be the random graph with edge probability $1/n^\alpha$; we know that it satisfies the 0-1 law for first order logic. We deal with the failure of the 0-1 law for stronger logics: ${\mathbb {L}}_{\infty , \mathbf{k}}, \mathbf{k}$ a large enough natural number and the inductive logic.

This work was partially supported by European Research Council grant 338821. Publication 1061 on Shelah’s list. The author thanks Alice Leonhardt for the beautiful typing.