A note on the minimum degree of minimal Ramsey graphs

In this note, we briefly rectify oversights in the works of several authors on s_r (K_k), the Ramsey parameter introduced by Burr, Erdős and Lovász 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. We show that s_r (K_k+1) = O(k³ r³ ln³ k), improving the best known bounds when k ≥ 8 and k² ≤ r ≤ O(k⁴/ ln⁶ k).