Is Post's Problem undecidable for fixed number of tiles i.e. $n\geq5$

So, I'm refreshing my undecidablility knowledge, and I came across this statement reading about Post's Correspondence Problem on Wikipedia:


A simple variant is to fix n, the number of tiles. This problem is decidable if n ≤ 2, but remains undecidable for n ≥ 5. It is unknown whether the problem is decidable for 3 ≤ n ≤ 4.

I'm a little confused about how to parse this sentence formally.

Let $POST(n) = \\ \{((a_1, b_1),\ldots, (a_n, b_n) \mid a_i, b_i \in \{0,1\}^*, \\ \exists j_1 \ldots j_m \ldotp a_{j_1} \cdots a_{j_m} = b_{j_1} \cdots b_{j_m} \}$

Which of the following is true?

• $\forall n \ldotp n \geq 5 \implies POST(n)$ is undecidable

• $\bigcup_{n \geq 5} POST(n)$ is undecidable

i.e. if I'm showing that a problem is undecidable, can I show that that problem can be used to solve any 5-tile PCP instance? Or am I not allowed to make such assumptions on the number of tiles?

It is the first version: $\forall n,\,n\geq 5\Rightarrow\,POST(n)$ is undecidable. The reason for the number $5$ is explained in the article of Turlough Neary cited in wikipedia page. The case $n=2$ is proven in

A. EHXENFEUCHT, J. KARHUMAKI, G. ROZENBERG, "THE (GENERALIZED) POST CORRESPONDENCE PROBLEM WITH LISTS CONSISTING OF TWO WORDS IS DECIDABLE",
Theoretical Computer Science 21 (1982) 119-144.

StackExchange Computer Science Q#71502, answer score: 4