An Erdős--Szekeres type result for words with repeats

We prove an Erdős--Szekeres type result for finite words over N with repeated values. Specifically, we define a repeat in a word to be an occurrence of a value which is not its first occurrence. We define an occurrence of a pattern $π$ in a word w to be a (not necessarily consecutive) subword of w that is order isomorphic to $π$. In this note, we show that every word with kn⁶+1 repeats contains one of the following patterns: 0ᵏ⁺², 0011⋯ nn, nn⋯1100, 012 ⋯ n012 ⋯ n, 012 ⋯ nn⋯ 210, n⋯ 210012⋯ n, n⋯ 210n⋯ 210. Moreover, when $k=1$, we show that this is best possible by constructing a word with n⁶ repeats that does not contain any of these patterns. 11 pages, 9 figures