TY - JOUR
T1 - On the Probability of Independent Sets in Random Graphs
AU - Krivelevich, Michael
AU - Sudakov, Benny
AU - Vu, Van H.
AU - Wormald, Nicholas C.
PY - 2003/1
Y1 - 2003/1
N2 - Let k be the asymptotic value of the independence number of the random graph G(n, p). We prove that if the edge probability p(n) satisfies p(n) ≫ n-2/5ln6/5n then the probability that G(n, p) does not contain an independent set of size k - c, for some absolute constant c > 0, is at most exp{ -cn2/(k4p)}. We also show that the obtained exponent is tight up to logarithmic factors, and apply our result to obtain new bounds on the choice number of random graphs. We also discuss a general setting where our approach can be applied to provide an exponential bound on the probability of certain events in product probability spaces.
AB - Let k be the asymptotic value of the independence number of the random graph G(n, p). We prove that if the edge probability p(n) satisfies p(n) ≫ n-2/5ln6/5n then the probability that G(n, p) does not contain an independent set of size k - c, for some absolute constant c > 0, is at most exp{ -cn2/(k4p)}. We also show that the obtained exponent is tight up to logarithmic factors, and apply our result to obtain new bounds on the choice number of random graphs. We also discuss a general setting where our approach can be applied to provide an exponential bound on the probability of certain events in product probability spaces.
UR - http://www.scopus.com/inward/record.url?scp=0037234549&partnerID=8YFLogxK
U2 - 10.1002/rsa.10063
DO - 10.1002/rsa.10063
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AN - SCOPUS:0037234549
SN - 1042-9832
VL - 22
SP - 1
EP - 14
JO - Random Structures and Algorithms
JF - Random Structures and Algorithms
IS - 1
ER -