An Erd\H{o}s--Szekeres type result for words with repeats

TL;DR

This paper extends Erdős–Szekeres type results to finite words over natural numbers with repeated values, identifying specific patterns guaranteed in words with sufficiently many repeats.

Contribution

It establishes a new combinatorial result for words with repeats, showing that large enough words contain certain patterns, and proves optimality for a specific case.

Findings

Words with kn^6+1 repeats contain specific patterns.

Constructed words with n^6 repeats avoid these patterns.

The result is tight for the case k=1.

Abstract

We prove an Erd\H{o}s--Szekeres type result for finite words over $\mathbb{N}$ with repeated values. Specifically, we define a \emph{repeat} in a word to be an occurrence of a value which is not its first occurrence. We define an occurrence of a \emph{pattern} $\pi$ in a word $w$ to be a (not necessarily consecutive) subword of $w$ that is order isomorphic to $\pi$. In this note, 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.