Failure of 0-1 law for sparse random graph in strong logics (Sh1062)

Saharon Shelah*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

This work was partially supported by European Research Council grant 338821. Publication 1062 on Shelah’s list. The author thanks Alice Leonhardt for the beautiful typing. 3.0(A) The Question. Let Gn, p be the random graph with set of nodes [n] = {1, …, n}, each edge of probability p ∈ [0, 1]R, the edges being drawn independently, see 2 below. On 0-1 laws (and random graphs) see Spencer [Spe01] or Alon and Spencer [AS08], in particular on the behaviour of the random graph Gn,1/nα for α ∈ (0, 1)R irrational. On finite model theory see Flum and Ebbinghaus [EF06], e.g., on the logic L∞, k (see 1) and on LFP (least fixed point1) logic. A characteristic example of what can be expressed by it is “in the graph G there is a path from the node x to node y”; this is close to what we use. We know that Gn, p, i.e., p constant satisfies the 0-1 law for first order logic (proved independently by Fagin [Fag76] and Glebskii et al. [GKLT69]). This holds also for many stronger logics like L∞, k and LFP logic. If α ∈ (0, 1)R is irrational, the 0-1 law holds for Gn,(1/nα) and first order logic, see, e.g., [AS08].

Original languageEnglish
Title of host publicationBeyond First Order Model Theory
PublisherCRC Press
Pages77-102
Number of pages26
ISBN (Electronic)9781498754019
ISBN (Print)9781498753975
DOIs
StatePublished - 1 Jan 2017

Bibliographical note

Publisher Copyright:
© 2017 by Taylor and Francis Group, LLC.

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