Ella Puerto

math.CO — Combinatorics

Iterating the Gupta–Ndiaye–Norin–Wei method: a computer-verified bound $R(k)\le 3.7698^{k+o(k)}$

Contributed by Ella Puerto

Gupta, Ndiaye, Norin and Wei (GNNW) recast the Campos–Griffiths–Morris–Sahasrabudhe method as an inductive theorem that turns a known bound on off-diagonal Ramsey numbers into a better one. With two applications they obtained $R(k)\le (3.7992\ldots)^{k+o(k)}$. They also reported an unverified, AI-suggested third application giving $3.78233\ldots$. We make three observations about their framework. (i) The free function $M$ can be optimised pointwise. (ii) For a fixed input, the best output exponent is the least solution of an explicit first-order differential inequality. (iii) GNNW's Theorem 14 needs only $F\in C^1$ with locally Lipschitz derivative. We then construct an explicit chain of eight applications and verify every hypothesis by interval arithmetic on the whole range $0<\lambda\le 1$, with a hand-proved estimate covering $\lambda\to0$. The verification was done twice, with independent code, libraries and enclosure strategies. This proves $R(k)\le 3.76973^{k+o(k)}$, and in particular confirms GNNW's $3.78233$. Numerically, iterating the optimal step converges to about $3.7690$, which suggests this is close to the limit of the GNNW method in its present form.