A note on Erd\"os-Faber-Lov\'asz Conjecture and edge coloring of complete graphs

A linear hypergraph is intersecting if any two different edges have exactly one common vertex and an $n$-quasicluster is an intersecting linear hypergraph with $n$ edges each one containing at most $n$ vertices and every vertex is contained in at least two edges. The Erd\"os-Faber-Lov\'asz Conjecture states that the chromatic number of any $n$-quasicluster is at most $n$. In the present note we prove the correctness of the conjecture for a new infinite class of $n$-quasiclusters using a specific edge coloring of the complete graph.