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On a Conjecture of Erdös, Faber, and Lovász about n-Colorings

  • Neil Hindman (a1)
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Let be a finite family of finite sets with the property that |AB| ≦ 1 whenever A and B are distinct members of . It is a conjecture of Erdös, Faber, and Lovász ([1] a 50 pound problem and [2] a 100 dollar problem) that there is an n-coloring of (i.e., a function such that AB = ∅ whenever A and B are distinct members of with f(A) = f(B). They actually state the conjecture in a different form. They actually state the conjecture in a different form. Namely, if n is a positive integer and is a family of n sets satisfying (1) |Bp | = n for each p and (2) |Bp Bq | ≦ 1 when pq, then there is an n-coloring of the elements of so that each set Bp gets all n colors. The equivalence of the two forms is easily seen by interchanging the roles of elements and sets.

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References
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1. Erdos, P., Problems and results in graph theory and combinatorial analysis, Proceedings of the Fifth British Combinatorial Conference, 169-92. Congressus Numerantium, No. XV, Utilitas Math. (1976).
2. Erdos, P., Some recent problems and results on graph theory, combinatorics and number theory, Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory and Computing, 3-14. Congressus Numerantium, No. XVII, Utilitas Math. (1976).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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