4-tangrams are 4-avoidable

TL;DR

This paper investigates the minimal alphabet size needed for infinite words to avoid certain tangram patterns with limited cuts, establishing that four-letter alphabets are necessary for avoiding tangrams with cut number four.

Contribution

It proves that the minimal alphabet size for avoiding tangrams with cut number four is four, resolving a previously open question.

Findings

For cut number one and two, the minimal alphabet size is three.

For cut number three and four, the minimal alphabet size is four.

Answers a question posed by D extk{e}bski et al. about tangram avoidance.

Abstract

A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one corresponds to squares. For $k\ge1$, let $t(k)$ denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most~$k$. The existence of infinite ternary square-free words shows that $t(1)=t(2)=3$. We show that $t(3)=t(4)=4$, answering a question from D\k{e}bski, Grytczuk, Pawlik, Przyby\l{}o, and \'Sleszy\'nska-Nowak.