"Exactly k" is not clear because the picture will still fall if you remove k+1 nails. I think it's harder to get clearer than the way it's stated in the paper:

"Exactly k" is not clear because the picture will still fall if you remove k+1 nails. I think it's harder to get clearer than the way it's stated in the paper: "the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed."