  We prove an upper bound for characters of the symmetric groups. Namely, we
show that there exists a constant a>0 with a property that for every Young
diagram \lambda with n boxes, r(\lambda) rows and c(\lambda) columns |Tr
\rho^\lambda(\pi) / Tr \rho^\lambda(e)| < [a max(r(\lambda)/n,
c(\lambda)/n,|\pi|/n) ]^{|\pi|}, where |\pi| is the minimal number of factors
needed to write \pi\in S_n as a product of transpositions. We also give uniform
estimates for the error term in the Vershik-Kerov's and Biane's character
formulas and give a new formula for free cumulants of the transition measure.
