Graphs with asymmetric Ramsey properties

Given positive integers $k$ and $\ell$ we write $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we denote by $R(k)$ the minimum $n$ such that $K_n\rightarrow (K_k,K_k)$. By using probabilistic methods and hypergraph containers we prove that for every integer $k \geq 3$, there exists a graph $G$ such that $G \nrightarrow (K_k,K_k)$ and $G \rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as a variation of a classical theorem of Ne\v{s}et\v{r}il and R\"odl [The Ramsey property for graphs with forbidden complete subgraphs, Journal of Combinatorial Theory, Series B, 20 (1976), 243-249], who proved that for every integer $k\geq 2$ there exists a graph $G$ with no copies of $K_k$ such that $G\rightarrow(K_{k-1}, K_{k-1})$.