Knights necessary to mate king on large finite board How many knights would be necessary to mate opponent's king on large finite boards like billion times billion squares? 

Are they fast enough to surround opponent's king in very limited constant count for various finite board sizes, or would you need some f(board size) of them, let's say millions for our board? Would there be a difference if attacking side could use it's king too?

It's not clear how geometry works here but it looks like that when king starts running, it's pretty fast and only two knights are fast enough to followhim. On the other hand, knights can build setups giving additional time for some distant colleague to join slowly. We start with king in the center and knights in one corner and 50 moves rule doesn't apply...
Not rigerous, however:


In lateral movement two knights move as fast as one king.
We know from end-game theory that two knights are not able to force the king into positions.


.. thus I'd say it's very unlikely that unlimited knights+king would be able to mate a king that has a free direction to run in (unlimited board). If it's possible to corner it however (limited board and unlimited moves), unlimited knights are certainly enough, as they could simply divide up the board and slowly decrease the available space.