TY - JOUR
T1 - Zero-one laws for graphs with edge probabilities decaying with distance. Part II
AU - Shelah, Saharon
PY - 2005
Y1 - 2005
N2 - Let Gnn 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 P|i-j| = 1/|i -j|α, 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.
AB - Let Gnn 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 P|i-j| = 1/|i -j|α, 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.
UR - http://www.scopus.com/inward/record.url?scp=22544448964&partnerID=8YFLogxK
U2 - 10.4064/fm185-3-2
DO - 10.4064/fm185-3-2
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AN - SCOPUS:22544448964
SN - 0016-2736
VL - 185
SP - 211
EP - 245
JO - Fundamenta Mathematicae
JF - Fundamenta Mathematicae
IS - 3
ER -