Subexponential estimates in Shirshov's theorem on height

Abstract

Suppose that is a free -generated associative ring with the identity . In 1993 Zelmanov put the following question: is it true that the nilpotency degree of has exponential growth?We give the definitive answer to Zelmanov's question by showing that the nilpotency class of an -generated associative algebra with the identity is smaller than , where This result is a consequence of the following fact based on combinatorics of words. Let , and be positive integers. Then all words over an alphabet of cardinality whose length is not less than are either -divisible or contain ; a word is -divisible if it can be represented in the form so that 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 -divisible words over an alphabet of cardinality has height over the set of words of degree , where Bibliography: 40 titles.