  We prove that $s_r(K_{k+1}) = O(k^3 r^3 \log^3 k)$, where $s_r(K_k)$ is the
Ramsey parameter introduced by Burr, Erd\H{o}s and Lov\'{a}sz in 1976, which is
defined as the smallest minimum degree of a graph $G$ such that any
$r$-colouring of the edges of $G$ contains a monochromatic $K_k$, whereas no
proper subgraph of $G$ has this property.
