Belov, Mikhail Kharitonov'
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    Subexponential estimates in Shirshov theorem on height
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[**Keywords:**]{} Shirshov theorem on height, word combinatorics, $n$-divisibility, Dilworth theorem, Burnside-type problems.

Introduction
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Shirshov theorem on height
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In 1958 A. I. Shirshov has proved his famous theorem on height ([@Shirshov1], [@Shirshov2]).

A word $W$ is called [*$n$-divisible*]{} if $W$ can be represented in the form $W = vu_1u_2\cdots u_n$ such that $u_1\succ
u_2\succ\dots\succ u_n$.

In this case any non-identical permutation $\sigma$ of subwords $u_i$ produces a word $W_\sigma =
vu_{\sigma(1)}u_{\sigma(2)}\cdots u_{\sigma(n)}$, which is lexicographically smaller than $W$. Some authors take this feature as the definition of $n$-divisibility.

A -algebra $A$ is called an algebra [*of bounded height $h=\Ht_Y(A)$ over a set of words $Y = \{ u_1, u_2,\ldots\}$*]{} if $h$ is the minimal integer such that any word $x$ from $A$ can be represented in the form $$x = \sum_i \alpha_i u_{j_{(i,1)}}^{k_{(i,1)}}
u_{j_{(i, 2)}}^{k_{(i,2)}}\cdots
u_{j_{(i,r_i)}}^{k_{(i,r_i)}}$$ where $\{r_i\}$ don’t exceed $h$. The set $Y$ is called [*a Shirshov basis*]{} for $A$. If no misunderstanding can occur, we use $h$ instead of $\Ht_Y(A)$.

([@Shirshov1], [@Shirshov2]) [*The set of not $n$-divisible words in a finitely generated algebra with an admissible polynomial identity has bounded height $H$ 