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

The paper is focused on two-sided estimates for the essential height in Shirshov’s Height Theorem. The concepts of the selective height and strong n-divisibility directly related to the height and n-divisibility are introduced. We prove lower and upper bounds for the selective height over nonstrongly n-divisible words of length 2. For any n and sufficiently large l these bounds differ at most twice. The case of words of length 3 is also studied. The case of words of length 2 can be generalized to the proof of an upper exponential estimate in Shirshov’s Height Theorem. The proof uses the idea of V.N. Latyshev related to the application of Dilworth’s theorem to the study of non n-divisible words.