Let Gₙ be the random graph on [n] = 1,...,n with the probability of i,j being an edge decaying as a power of the distance, specifically the probability being , where the constant α ∈ (0,1) is irrational. We analyze this theory using an appropriate weight function on a pair (A,B) of graphs and using an equivalence relation on B∖A. We then investigate the model theory of this theory, including a “finite compactness”. Lastly, as a consequence, we prove that the zero-one law (for first order logic) holds.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm185-3-2,
author = {Saharon Shelah},
title = {Zero-one laws for graphs with edge probabilities decaying with distance. Part II},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {211-245},
zbl = {1077.03016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm185-3-2}
}
Saharon Shelah. Zero-one laws for graphs with edge probabilities decaying with distance. Part II. Fundamenta Mathematicae, Tome 185 (2005) pp. 211-245. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm185-3-2/