The Number of Threshold Words on $n$ Letters Grows Exponentially for Every $n\geq 27$

TL;DR

This paper proves that for all sufficiently large alphabet sizes, the number of threshold words of a given length grows exponentially, confirming a significant part of Ochem's conjecture.

Contribution

It establishes exponential growth of threshold words for all alphabet sizes n ≥ 27, resolving nearly all cases of a longstanding conjecture.

Findings

Exponential growth of threshold words for n ≥ 27

Complete proof of all but finitely many cases of Ochem's conjecture

Advancement in understanding combinatorial properties of threshold words

Abstract

For every $n\geq 27$, we show that the number of $n/(n-1)^+$-free words (i.e., threshold words) of length $k$ on $n$ letters grows exponentially in $k$. This settles all but finitely many cases of a conjecture of Ochem.