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Statement

After the proof of Theorem 1 the paper says (p. 117): "It is clear that Theorem 1 is far short of what must be true. For instance, in view of (1), the value of $r(n,n)-r(n-1,n-1)$ must be exponentially large in $n$ on the average, and it seems almost certain that this difference has an exponential lower bound as well." Display (1) (p. 115) is

$$\sqrt2\cdot n\cdot2^{n/2}/e\ \le\ r(n,n)\ \le\ \binom{2n-2}{n-1},$$

quoted by the paper without proof or attribution (it points only to the survey [2]). The attributions are made here: the upper bound is Erdős and Szekeres's, and the lower bound is Spencer's asymptotic bound without its factor $1+o(1)$, so (1) as printed, for the paper's range $n\ge2$, fails at $n=2$, where $\sqrt2\cdot2\cdot2/e\approx2.08>r(2,2)=2$; the remark uses (1) only asymptotically. The first sentence about the average is immediate from the lower bound in (1); the second is an expectation, not a theorem.

Source. S. A. Burr, P. Erdős, R. J. Faudree and R. H. Schelp, On the difference between consecutive Ramsey numbers, Utilitas Mathematica 35 (1989), 115--118; the remark on printed p. 117 (PDF p. 3 of the scan) and display (1) on p. 115 (PDF p. 1), read on the page images.

Read depth. Claims checked: the remark and display (1) were read clause by clause on the page images.

Proof pointer

None; the remark is not a theorem.

Dependencies

Display (1), which the paper quotes from the literature.

Bears on

• Problem 1030: context only. The authors expect exponential gaps in the diagonal step $r(n,n)-r(n-1,n-1)$, not in the step from $R(k,k)$ to $R(k+1,k)$, and even an exponential lower bound on a gap would not by itself bound $R(k+1,k)/R(k,k)$ away from $1$, since $R(k,k)$ is itself exponential. The ratio conjecture predates the paper (display (7) of Erdős's 1981 survey, which credits it to Burr and Erdős), and the paper does not state it.
• Problem 812: the same expectation for the diagonal step $R(n)-R(n-1)$, whose average size is exponential by display (1); an exponential increment would answer that page's second question ($R(n+1)-R(n)\gg n^2$) but not by itself the first, which needs an increment of order $R(n)$, and the paper proves neither.