Subexponential estimates in Shirshov's theorem on height

Suppose that F_2,m is a free 2-generated associative ring with the identity x^m = 0. In 1993 Zelmanov put the following question: is it true that the nilpotency degree of F_2,m has exponential growth? We give the defnitive answer to Zelmanov's question by showing that the nilpotency class of an l-generated associative algebra with the identity x^d = 0 is smaller than Ψ(d,d,l), where (equation) This result is a consequence of the following fact based on combinatorics of words. Let l, n and d ≥ n be positive integers. Then all words over an alphabet of cardinality l whose length is not less than Ψ(n,d,l) are either n-divisible or contain x^d; a word W is n-divisible if it can be represented in the form W = W ₀W₁ · · · W_n so that W₁,., W_n are placed in lexicographically decreasing order. Our proof uses Dilworth's theorem (according to V. N. Latyshev's idea). We show that the set of not n-divisible words over an alphabet of cardinality l has height h < Φ(n, l) over the set of words of degree ≤ n-1, where 87 12 logc, n+48 Φ(n, l) = 2⁸⁷l · n^12log ₃n+48 Bibliography: 40 titles.