Monadic second-order properties of very sparse random graphs

2016/09/05 by L. B. Ostrovsky, Ostrovsky, L. B., Maksim Zhukovskii +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1609.01102

openalex publication_date 2016/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study asymptotical probabilities of first order and monadic second order properties of Erdos-Renyi random graph G(n,n^-a). The random graph obeys FO (MSO) zero-one k-law if for any first order (monadic second order) formulae it is true for G(n,n^-a) with probability tending to 0 or to 1. Zero-one k-laws are well studied only for the first order language and a < 1. We obtain new zero-one k-laws (both for first order and monadic second order languages) when a > 1. Proofs of these results are based on the existed study of first order equivalence classes and our study of monadic second order equivalence classes. The respective results are of interest by themselves.

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