Problem 812

Is it true that

\frac{R(n+1)}{R(n)}\geq 1+c

for some constant $c>0$ , for all large $n$ ? Is it true that

R(n+1)-R(n) \gg n^2?

Source: erdosproblems.com/812

Graph theory
Ramsey theory

Wiki page

Status

Open

No claim settles this problem.

Open: the site labels the problem OPEN, and the only source results in hand are additive: the Corollary of Burr, Erdős, Faudree and Schelp specializes on the diagonal to $R(n+1)-R(n)\ge4n-4$ for $n\ge2$ (the site's commentary prints $4n-8$ ), and with the lower bound for $R(3,k)$ their Theorem 2 gives the two-step bound $R(n+2)-R(n)\gg n^2/\log n$ . Nothing in hand bounds $R(n+1)/R(n)$ away from $1$ , proves $R(n+1)-R(n)\gg n^2$ , or refutes either question, and no source doing so was found in the search whose scope the Current assessment records. This is a bounded negative finding, not a certificate of openness.

BEFS89: Burr, S. A., Erdős, P., Faudree, R. J. and Schelp, R. H., On the difference between consecutive Ramsey numbers. Utilitas Math. 35 (1989), 115--118. Theorem 1 and Corollary, p. 115; Theorem 2, p. 116; the remark on growth, p. 117. Library home: Burr et al. 1989.
Er91: Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988), Wiley (1991), 397--406. The site's source key; not held, recorded without an access attempt.
Er88: Erdős, P., Problems and results in combinatorial analysis and graph theory. Discrete Math. 72 (1988), 81--92; displays (5) and (6) on p. 83 and their comment on p. 84. Library home: Erdős 1988.
Er81c: Erdős, P., Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (Calcutta, 1980), Lecture Notes in Math. 885 (1981), 9--17; displays (7) and (8) on p. 11. Library home: Erdős 1981.
Ki95: Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\log t$ . Random Structures Algorithms 7 (1995), 173--207; the lower bound $R(3,t)\gg t^2/\log t$ used in the two-step derivation below, compiled on the page of Problem 165. Library home: Kim 1995.
Mo26: Morris, R., Some recent results in Ramsey theory. Proceedings of the International Congress of Mathematicians 2026, Vol. 2, 210--239, DOI 10.1137/25m1833369; arXiv:2601.05221v1 (8 January 2026). Consulted for coverage; it contains nothing on consecutive differences. Library home: Morris 2026.

Problem 812

Status

Proof claims

Discussion