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