Zero-one laws for graphs with edge probabilities decaying with distance. Part I

Saharon Shelah
2002 Fundamenta Mathematicae  
Let G n be the random graph on [n] = {1, . . . , n} with the possible edge {i, j} having probability p |i−j| = 1/|i − j| α for j = i, i + 1, i − 1 with α ∈ (0, 1) irrational. We prove that the zero-one law (for first order logic) holds. |l − m| α p |{l : (l,l+1) is an edge}| 1 ×(1 − p 1 ) |{l : (l,l+1) is not an edge}| where l, m ≤ k and J 1 = {{l, m} : (l, m) is an edge and |l − m| > 1}, J 2 = {{l, m} : (l, m) is not an edge and |l − m| > 1}. Hence the probability that for no i < n/k does the
more » ... o i < n/k does the mapping l → ki + l embed H into M n is Hence if βk α( k 2 ) = n/k, that is, β = n/k α( k 2 )+1 then this probability is ≤ e −β . This is because e −β ∼ (1 − β/n) n . We obtain .
doi:10.4064/fm175-3-1 fatcat:7dpbpvklsnernbs2oxdupp2p4e