A note on edge-colourings avoiding rainbow \(K_{4}\) and monochromatic \(K_{m}\) (Q2380208)

Summary: We study the mixed Ramsey number \(\max R(n, K_m, K_r)\), 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 \(\max R(n, K_m, K_4) \leq n^{3/2} \sqrt{2m}\) for all \(m \geq 3\). Further, we discuss a possible way to improve their lower bound on \(\max R(n, K_4, K_4)\) based on incidence graphs of finite projective planes.