Upper Bounds for Sequence Saturation

TL;DR

This paper investigates the saturation function for sequences, proving linear bounds for various sequence types, and introduces an algorithm and linear program to analyze and compute these bounds.

Contribution

It provides new bounds and algorithms for the sequence saturation problem, extending previous results to broader classes of sequences and offering a method to compute exact saturation values.

Findings

Proved $ ext{Sat}(n,u)=O(n)$ for all sequences $u$ in several families.

Developed an algorithm to construct $u$-saturated sequences.

Created a linear program to compute $ ext{Sat}(n,u)$ exactly.

Abstract

In this paper, we study the saturation function $\mathrm{Sat}(n,u)$ for sequences. Saturation for sequences was introduced by Anand, Geneson, Kaustav, and Tsai (2021), who proved that $\mathrm{Sat}(n,u)=O(n)$ for two-letter sequences $u$ and conjectured that this bound holds for all sequences. We present an algorithm that constructs a $u$-saturated sequence on $n$ letters and apply it to show $\mathrm{Sat}(n,u)=O(n)$ for several families of sequences $u$, including all repetitions of the form $abcabc\dots$. We further establish $\mathrm{Sat}(n,u)=O(n)$ for a broad class of sequences of the form $aa\dots bb$. In addition, we prove that for most sequences $u$, there exists an infinite $u$-saturated sequence. For three-letter sequences of the form $abc\dots xyz$, where $a,b,c$ are distinct and $xyz$ is a permutation of $abc$, we show -- under certain structural assumptions on $u$ -- that $\mathrm{Sat}(n,u)=O(n)$. Finally, we describe a linear program that computes the exact value of $\mathrm{Sat}(n,u)$ for arbitrary $n$ and $u$.