  We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum
number of colours in an edge-colouring of the complete graph K_n, such that K_n
has no monochromatic complete subgraph on m vertices and no rainbow complete
subgraph on r vertices. Improving an upper bound of Axenovich and Iverson, we
show that maxR(n,K_m,K_4) <= n^{3/2}\sqrt{2m} for all m >= 3. Further, we
discuss a possible way to improve their lower bound on maxR(n,K_4,K_4) based on
incidence graphs of finite projective planes.
