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On a Diagonal Conjecture for Classical Ramsey Numbers
- Publication Year :
- 2018
-
Abstract
- Let $R(k_1, \cdots, k_r)$ denote the classical $r$-color Ramsey number for integers $k_i \ge 2$. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if $k_1, \cdots, k_r$ are integers no smaller than 3 and $k_{r-1} \leq k_r$, then $R(k_1, \cdots, k_{r-2}, k_{r-1}-1, k_r +1) \leq R(k_1, \cdots, k_r)$. We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems. Let $R_r(k)$ stand for the $r$-color Ramsey number $R(k, \cdots, k)$. It is known that $\lim_{r \rightarrow \infty} R_r(3)^{1/r}$ exists, either finite or infinite, the latter conjectured by Erd\H{o}s. This limit is related to the Shannon capacity of complements of $K_3$-free graphs. We prove that if DC holds, and $\lim_{r \rightarrow \infty} R_r(3)^{1/r}$ is finite, then $\lim_{r \rightarrow \infty} R_r(k)^{1/r}$ is finite for every integer $k \geq 3$.<br />Comment: 10 pages
- Subjects :
- Mathematics - Combinatorics
05C55, 05C35
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1810.11386
- Document Type :
- Working Paper