An Erdős--Szekeres type result for words with repeats

TL;DR

The paper extends Erdős--Szekeres-type ideas to words over $\,\mathbb{N}$ with repeats, aiming to identify unavoidable patterns when the number of repeats is large. It uses Erdős--Szekeres to extract monotone subsequences from carefully chosen subwords and classifies unavoidable patterns into three families: $0^{k+2}$, $(0011\cdots nn)^e$ for $e\in\{id,rev\}$, and $(012 \cdots n)^{e_1}(012 \cdots n)^{e_2}$ for $e_1,e_2\in\{id,rev\}$. The main result shows that any word with $k n^6+1$ repeats contains one of these patterns, with the bound proven tight for $k=1$ via an explicit extremal construction. The work lays groundwork for understanding inversion sequences and raises natural questions for $k>1$, such as optimal repeat thresholds and extensions to more restricted word classes.

Abstract

In this short note, we prove an Erdős--Szekeres type result for words with repeats. Specifically, we show that every word with $kn^6+1$ repeats contains one of the following patterns: $0^{k+2}$, $0011\cdots nn$, $nn\cdots1100$, $012 \cdots n012 \cdots n$, $012 \cdots nn\cdots 210$, $n\cdots 210012\cdots n$, $n\cdots 210n\cdots 210$. Moreover, when $k=1$, we show that this is best possible by constructing a word with $n^6$ repeats that does not contain any of these patterns.

Figures (9)

• Figure 1: The word 13043134.
• Figure 2: An occurrence of a pattern type \ref{['2']} in $w'$.
• Figure 3: The direct sum and skew sum of two words.
• Figure 4: The word $r$ formed by taking the skew sums of $n$ copies of the word $t=12\cdots n$.
• Figure 5: The word $r'=(r^{\ominus n})^{\oplus n}$.

Theorems & Definitions (10)

• Theorem 1.1: Erdős--Szekeres Theorem Erdos1935
• Theorem 1.2
• Proposition 1.3
• proof : Proof of Theorem \ref{['prop: word with m repeats']}
• Claim 2.1
• proof : Proof of \ref{['claim:main']}
• Lemma 3.1
• proof
• proof : Proof of Proposition \ref{['prop: bound is tight']}
• Proposition 4.2