A question about red

I've been working merrily along on this year's ornament for my daughter - a design by The Sunflower Seed from (I think) the 2014 JCS ornament issue.  I've looked all over for a picture of it finished, and have only found one, via Pinterest:

(From a French blog - not my stitching)

However, even this stitcher made changes.  Which brings me to my question - do all reds (particularly hand-dyeds) fuzz and/or bleed?  Because this design is supposed to have a border of red, then white, then green.  I did the red (which the stitcher of the piece pictured did not), and as I was stitching the white, it started to look.....positively pink.  I thought it was going to be okay, but the more I stitched the less I liked it.  So I frogged out all the white, and decided to leave it with just the red border - I might make a white and green twisted cording when I do the final finishing.

I've also started this little freebie from Hands On Design to be a gift for a friend - I'm leaving off the bottom of the quote about stitching, since he doesn't stitch.  But now I keep humming that darned Styx song.... you're welcome.  >:)

I should be back with a picture or 2 of my own soon!