Proper edge colorings of Cartesian products with rainbow \(C_4\)-s (Q6133668)

The famous Brown-Erdős-Sós conjecture (BES-conjecture) assert that if a \(3\)-uniform hypergraph (\(3\)-graph) \(H\) on \(n\) vertices does not contain \(k\) edges on at most \(k+3\) vertices then it has \(o(n^{2})\) edges [\textit{W. G. Brown} et al., in: F. Harary (ed.), New directions in the theory of graphs. Proceedings of the third Ann Arbor conference on graph theory. New York, London: Academic Press. 53--63 (1973; Zbl 0258.05132)]. \textit{I. Z. Ruzsa} and \textit{E. Szemerédi} [Colloq. Math. Soc. János Bolyai 18, 939--945 (1978; Zbl 0393.05031)] stated that if a \(3\)-uniform hypergraph (or shortly \(3\)-graph) \(H\) on \(n\) vertices does not contain \(3\) edges on at most \(6\) vertices then it has \(o(n^{2})\) edges. Moreover, the \((7, 4)\)-problem is a spectral case of the BES conjecture and it claims that: Conjecture. Assume that a \(3\)-graph \(H\) on \(n\) vertices does not contain \(4\) edges on at most \(7\) vertices (briefly we say that \(H\) is \((7, 4)\)-free). Then \(|e(H)|=o(n^{2})\). An edge coloring of a graph \(G\) is called a rainbow coloring if the edges of every quadrangle (\(C_{4}\)) of \(G\) are colored with distinct colors. An edge coloring of a graph \(G\) is proper if incident edges of \(G\) must receive different colors. Based on these, the authors define proper rainbow coloring as \(B\)-coloring and write \(q_{B}(G)\) for the smallest number of colors needed for a \(B\)-coloring of a graph \(G\). In this paper, the authors consider \(q_{B}(G)\) for Cartesian products of paths and cycles. Their main tool is a lemma which provides that \(q_{B}(G\square H) \leq q_{B}(G) + q_{B}(H)\) if \(\chi(G) \leq q_{B}(H)\), \(\chi(H) \leq q_{B}(G)\), where \(\chi(G)\) is the chromatic number of \(G\). Furthermore, \(B\)-coloring is similar to the well-studied concept of star-edge colorings, which is also related to the \((7,4)\)-problem. So the results of this paper provide a positive answer to the \((7,4)\)-problem in some sense. All the certification procedures are strict, and the ways of proof are properly novel.