Is Post's Correspondence Problem decidable with fixed word size?
For all $m \geq 3$, the problem is undecidable.

Proof by reduction from the word problem of unrestricted grammars:


Take an arbitrary formal grammar. W.l.o.g. all left and right sides of rules have length at most $3$.

This can be seen by translating any grammar into an equivalent TM and then converting back.
Map the resulting grammar to PCP instances; no tile is longer than the longest left or right side of a rule.

That is, with step 1, all tiles have length $\geq 3$.