arXiv:1207.2987v1 [math.RA] 12 Jul 2012 16R+05E УДК 512.5+512.64+519.1 Subexponential estimates in Shirshov theorem on height Alexei Belov, Mikhail Kharitonov July 13, 2012 Abstract In 1993 E. I. Zelmanov has put the following question in Dniester Notebook:Suppose that F 2,m is a free 2-generated associative ring with the identity x m =0. Is it true that the nilpotency degree of F 2,m has exponential growth? We give the definitive answer to E. I. Zelmanov’s question 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 Ψ(n, d, l)=2 18 l(nd) 3 log 3 (nd)+13 d 2 . This result is a consequence of the following fact based on combinatorics of words. Let l, n и 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 d-th power of a subword; a word W is n-divisible if it can be represented in the form W = W 0 W 1 ··· W n such that W 1 ,W 2 ,...,W n are placed in lexicographically decreasing order. Our proof uses Dilworth 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 Φ(n, l)=2 87 l · n 12 log 3 n+48 . Keywords: Shirshov theorem on height, word combinatorics, n-divisibility, Dilworth theorem, Burnside-type problems. 1 Introduction 1.1 Shirshov theorem on height In 1958 A. I. Shirshov has proved his famous theorem on height ([1], [2]). 1