Two-sided estimates of essential height in the Shirshov height theorem / *M. I. Kharitonov.* // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2012. № 2. P. 20-24
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 0].

The paper is focused on two-sided estimates of the essential height in Shirshov's Height theorem. The notions of the *selective height* and *strong n-divisibility* directly related to the *height and n-divisibility are introduced in the paper. We find lower and upper bounds for the selective height of non-strongly n-divided words over the words of length 2. These bounds differ by not more than twice for any n and sufficiently large l. The case of words of length 3 is also studied. The case of words of length 2 can be generalized to the proof of a subexponential estimate in Shirshov's Height theorem. The proof uses the idea of Latyshev related to the use of Dilworth's theorem to the of non-n-divided words.*

*Key words*:
essential height, Shirshov's height theorem, combinatorics of words, *n*-divisibility, Dilworth's theorem.