Balanced edge-colorings avoiding rainbow cliques of size four (Q6133158)

A balanced edge-coloring of the complete graph \(K_n\) is a coloring of the edges of \(K_n\) such that every vertex is incident to each color the same number of times. Let \(F\) be a given graph. We say that a balanced edge-coloring of \(K_n\) contains a rainbow \(F\) if \(K_n\) contains a subgraph isomorphic to \(F\) such that all of its edges are assigned different colors. The question of the existence of a rainbow \(F\) becomes interesting when one imposes a restriction on the number of colors used in the edge coloring. In [Ann. Discrete Math. 55, 81--88 (1993; Zbl 0791.05037)], \textit{P. Erdős } and \textit{Z. Tuza} asked whether for any given graph \(F\) one could choose \(n\) sufficiently large so that every balanced edge-coloring \(K_n\) with \(|E(F)|\) colors would contain a rainbow copy of \(F\). In this article, the authors settle this question in the negative for \(F=K_4\), by proving that for each \(k\geq 1\), there exists a balanced edge-coloring of \(K_{13^k}\) with 6 colors with no rainbow \(K_4\). The authors provide a specific balanced edge-coloring for \(K_{13}\) with this property. The existence of a balanced edge-coloring of \(K_{13^k}\) with 6 colors with no rainbow \(K_4\) for \(k>1\), follows from a result of \textit{M. Axenovich} and \textit{F. C. Clemen} in [``Rainbow subgraphs in edge-colored complete graphs'', Preprint, \url{arXiv:2209.13867}].