TY - JOUR
T1 - Every monotone graph property has a sharp threshold
AU - Friedgut, Ehud
AU - Kalai, Gil
PY - 1996
Y1 - 1996
N2 - In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let Vn(p) = {0, 1}n denote the Hamming space endowed with the probability measure μp defined by μp(ε1,ε2,...,εn) = pk · (1-p)n-k, where k = ε1 + ε2 + ⋯ + εn. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on {1,2,...,n} such that A is invariant under Γ. Theorem. For every symmetric monotone A, if μp(A) > ε then μq(A) > 1-ε for q = p + c1 log(1/2ε)/log n. (c1 is an absolute constant.)
AB - In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let Vn(p) = {0, 1}n denote the Hamming space endowed with the probability measure μp defined by μp(ε1,ε2,...,εn) = pk · (1-p)n-k, where k = ε1 + ε2 + ⋯ + εn. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on {1,2,...,n} such that A is invariant under Γ. Theorem. For every symmetric monotone A, if μp(A) > ε then μq(A) > 1-ε for q = p + c1 log(1/2ε)/log n. (c1 is an absolute constant.)
UR - http://www.scopus.com/inward/record.url?scp=13744252028&partnerID=8YFLogxK
U2 - 10.1090/s0002-9939-96-03732-x
DO - 10.1090/s0002-9939-96-03732-x
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AN - SCOPUS:13744252028
SN - 0002-9939
VL - 124
SP - 2993
EP - 3002
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 10
ER -