# Cayley graph

## 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

## Cores of Cayley graphs ★★

Author(s): Samal

\begin{conjecture} Let $M$ be an abelian group. Is the \Def[core]{core (graph theory)} of a \Def{Cayley graph} (on some power of $M$) a Cayley graph (on some power of $M$)? \end{conjecture}

Keywords: Cayley graph; core