Ramsey number

Ramsey properties of Cayley graphs ★★★

Author(s): Alon

\begin{conjecture} There exists a fixed constant $c$ so that every abelian group $G$ has a subset $S \subseteq G$ with $-S = S$ so that the \Def[Cayley graph]{cayley graph} ${\mathit Cayley}(G,S)$ has no clique or independent set of size $> c \log |G|$. \end{conjecture}

Keywords: Cayley graph; Ramsey number

Diagonal Ramsey numbers ★★★★

Author(s): Erdos

Let $R(k,k)$ denote the $k^{th}$ diagonal \Def[Ramsey number]{ramsey number}.

\begin{conjecture} $\lim_{k \rightarrow \infty} R(k,k) ^{\frac{1}{k}}$ exists. \end{conjecture}

\begin{problem} Determine the limit in the above conjecture (assuming it exists). \end{problem}

Keywords: Ramsey number

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