Two-sided estimates for essential height in Shirshov's Height Theorem

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.