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Statement

The note added in proof (printed pp. 53--54) opens by remarking that work on Ramsey numbers had advanced since the paper was written, then reads (p. 53): "R. J. Faudree and R. H. Schelp [8] and, independently, V. Rosta [9], have shown that, except for $R(C_3,C_3)$ and $R(C_4,C_4)$,

$$R(C_m,C_n)=\begin{cases} 2n-1, & \text{for }3\le m\le n,\ m\text{ odd}\\ n+(m/2)-1, & 4\le m\le n,\ m,n\text{ even}\\ \max\{n+(m/2)-1,\,2n-1\text{ [sic]}\}, & 4\le m<n,\ m\text{ even},\ n\text{ odd."} \end{cases}$$

The maximum in the third case is printed with $2n-1$. Read that way it would always equal $2n-1$, since $m<n$, and at $m=4$, $n=5$ it would give $9$, against the paper's own $R(C_5,C_4)=7$ (p. 47); the reading $2m-1$ gives $7$ (a check made here).

The note continues on p. 54 with Faudree and Schelp's $R(P_m,P_n)=n+\lfloor(m+1)/2\rfloor$ for $1\le m\le n$ and their four-case formula for $R(C_m,P_n)$, "where $P_n$ is a path of length $n$", and records that T. D. Parsons evaluated $R(C_4,P_n)$ and $R(K_m,P_n)$. The note's references are Faudree and Schelp, "All Ramsey numbers for cycles in graphs", submitted to Discrete Mathematics; Rosta, submitted to J. Combinatorial Theory; and a personal communication of Parsons. Nothing in the note is proved in the paper.

The even case is the two-color value of the even-cycle problem: for $m=n=2k$ with $k\ge3$, $R(C_{2k},C_{2k})=3k-1$, which is the conjecture $R(C_{2n},C_{2n})=3n-1$ for $n>2$ that Section 4 (p. 53) draws from $R(C_6,C_6)=8$.

Source. J. A. Bondy and P. Erdős, Ramsey numbers for cycles in graphs, J. Combinatorial Theory Ser. B 14 (1973), 46--54; the note added in proof on printed pp. 53--54 (PDF pp. 8--9 of the scan), read on the page images at 130 dpi and, for the displayed formula, at 260 dpi; the text layer garbles the displays.

Read depth. Claims checked: the note's sentences and the three-case formula were read clause by clause on the page images. The paper gives no proof, and the papers of Faudree and Schelp and of Rosta are not held, so nothing is proof-checked.

Proof pointer

None in the paper; the note reports the results of its [8] and [9] without argument.

Dependencies

None stated; the results are Faudree and Schelp's and Rosta's.

Bears on

• Problem 555: the even case gives the two-color value $R_2(C_{2n})=3n-1$ for $n\ge3$, the $k=2$ entry of the problem's multicolor question; the odd and mixed cases are context.