Is the Erdős–Faber–Lovász conjecture open still? Is the Erdős–Faber–Lovász   Conjecture open still? According to Wikipedia it is unsolved still, but I think this is not hard to solve this conjecture.

Conjecture: If $n$ complete graphs, each having exactly $n$ vertices, have the property that every pair of complete graphs has at most one shared vertex, then the union of the graphs can be colored with $n$ colors.
My attempt: Consider the, $n-1$ complete graphs $K_n$ which all has a common vertex, say $u$. then this graph can be colored with $n$ colors. by choosing $n-1$ vertices (one from each graph $K_n\backslash \{u\}$) by different color and construct a graph with this vertices and a new vertex, say $v$. we can assign to $v$ color of $u$. Why this is not correct or maybe I don't understand the problem ?


On January 2021 The problem was settled for sufficiently large values of $n$ https://arxiv.org/abs/2101.04698
