TITLE: Does an existence of large cardinals have implications in number theory or combinatorics?
QUESTION [44 upvotes]: Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any interesting theorems in these areas that can be proved based on assumptions of existence of certain large cardinals? Or they are too far to change anything here?
I know that existence of certain cardinals implies consistency of some axiomatic systems of the set theory, which in turn can be expressed as a statement about certain huge Diophantine equations having no solutions, but it looks unlikely that we could use these Diophantine equations for anything else.

REPLY [12 votes]: Applications of Friedman's Jump-free theorem (the proof of which requires large cardinals) to distance functions in lattice graphs is on his website:  
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
It is item 11: Applications of Large Cardinals to Graph Theory, October 23, 1997.
This remarkable paper by Friedman arose when a homework problem presented in an upper division undergraduate course at UCSD-CSE (solvable by a version of Ramsey's theorem) was extended slightly and no one could solve it (including the instructor and other faculty members). In January 1997, Friedman gave a lecture at UCSD where he presented his jump-free theorem.  It was just what we needed to solve the homework problem!
Another extension of this homework problem has been shown by Friedman (in the above paper) to itself require large cardinals and is equivalent to the jump-free theorem.
These problems can be viewed as a "toy versions" of the string theory landscape or multiverse concept.
A fanciful presentation for high school level students showing how these simply stated but hard problems arise is at
http://cseweb.ucsd.edu/~gill/MultUnivSite/