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{\bf Veselin Jungi\'{c}, Tom\'{a}\v{s} Kaiser and Daniel Kr\'{a}l'}
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{\bf A Note on Edge-Colourings Avoiding Rainbow $K_4$ and Monochromatic $K_m$}
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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
$maxR(n,{K_m},{K_4}) \leq n^{3/2}\sqrt{2m}$ for all $m\geq
3$. 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.



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