We present a new method to show concentration of the upper tail of random variables that can be written as sums of variables with plenty of independence. We compare our method with the martingale method by Kim and Vu, which often leads to similar results.
Some applications are given to the number
X
G
of copies of a graph
G in the random graph
\Bbb G \Bbb G
(
n,p). In particular, for
G =
K
4 and
G =
C
4 we improve the earlier known upper bounds on —ln
\Bbb E\Bbb E
X
G
) in some range of
p =
p(
n).
Mathematics Subject Classification (2000):
60F05
* Research of the second author supported by KBN grants 2 P03A 032 16 and 2 P03A 015 23.