Subexponential estimations in Shirshov's height theorem

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In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that F₂, ₘ is a 2-generated associative ring with the identity xᵐ=0. Is it true, that the nilpotency degree of F₂, ₘ has exponential growth?" We show that the nilpotency degree of l-generated associative algebra with the identity xᵈ=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2¹⁸ l (nd)3 log₃ (nd)+13d². We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let l, n and d>n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Psi(n,d,l) are either n-divided or contain d-th power of subword, where a word W is n-divided, if it can be represented in the following form W=W₀ W₁...Wₙ such that W₁ >' W₂>'...>'Wₙ. The symbol >' means lexicographical order here. A. I. Shirshov proved that the set of non n-divided words over alphabet of cardinality l has bounded 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 was obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation. We show, that h<Phi(n,l), where Phi(n,l) = 2⁸⁷ n12 log₃ n + 48 l. Our proof uses Latyshev idea of Dilworth theorem application.