What is the fastest way you can think of to checkmate by only moving the king?

I believe @tpr's answers are the only ones to actually satisfy the requirements as stated. The side moving only the king must be the winning side.

Then moving the f-pawn makes no sense.

That is correct: the side wanting to checkmate with its king must start by moving either the d or the e pawn so as to deliver checkmate by discovered checkmate from either a bishop or the queen.

Probably I won’t get it ever ;)

I elaborated on @tpr ‘s nice idea and saved a whole move.

i don't know men, sorry :D

Here's an interesting one. It's similar to #14 but only uses pawn and king moves for both sides (with only 1 pawn move by the mating side). lichess.org/V0XskzCo#0

@Sarg0n @Chess_Dogg
Those are both good! Seven moves. Hard to see how they could be improved upon.

Dogg's little king dance while waiting for white to get into position is pleasing for some reason :-)

@tpr #12, #13 is not entirely true because mates are possible even when one starts with the f-pawn. Probably not within 7 moves. Example line starting with the f-pawn:

Opening lines for the attacker gives another possibilities as well:

Wow, nice lines! I did not even think of the h-file opening for discovered checkmate.