Geodesic
Two-sided estimates for essential height in Shirshov's Height Theorem
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2012), pp. 20-24

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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. 
@article{VMUMM_2012_2_a4,
     author = {M. I. Kharitonov},
     title = {Two-sided estimates for essential height in {Shirshov's} {Height} {Theorem}},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {20--24},
     publisher = {mathdoc},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2012_2_a4/}
}
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M. I. Kharitonov. Two-sided estimates for essential height in Shirshov's Height Theorem. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2012), pp. 20-24. http://geodesic.mathdoc.fr/item/VMUMM_2012_2_a4/