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In 1993 E. I. Zelmanov asked the following question in Dniester Notebook:"Suppose that F_{2, m} is a 2-generated associative ring with the identityx^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponentialgrowth?" We show that the nilpotency degree of l-generated associative algebrawith the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l(nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov bythis result. It is the consequence of one fact, which is based on combinatoricsof words. Let l, n and d>n be positive integers. Then all the words overalphabet of cardinality l which length is greater than Psi(n,d,l) are eithern-divided or contain d-th power of subword, where a word W is n-divided, if itcan be represented in the following form W=W_0 W_1...W_n such that W_1 >'W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshovproved that the set of non n-divided words over alphabet of cardinality l hasbounded height h over the set Y consisting of all the words of degree <n.Original Shirshov's estimation was just recursive, in 1982 double exponent wasobtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation.We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Ourproof uses Latyshev idea of Dilworth theorem application.

Subexponential estimations in Shirshov's height theorem (in English)