Erdős–Faber–Lovász Conjecture

In graph theory, the Erdős–Faber–Lovász conjecture is an unsolved problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972. It says:

If k complete graphs, each having exactly k 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 k colors.

Read more about Erdős–Faber–Lovász Conjecture:  Equivalent Formulations, History and Partial Results, Related Problems

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Erdős–Faber–Lovász Conjecture - Related Problems
... A version of the conjecture that uses the fractional chromatic number in place of the chromatic number is known to be true ... He shows that, for any fixed value of L, a finite calculation suffices to verify that the conjecture is true for all simple hypergraphs with that value of L ... Based on this idea, he shows that the conjecture is indeed true for all simple hypergraphs with L ≤ 10 ...

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