Geodesic
A partial generalization of the Livingstone-Wagner Theorem
Ars Mathematica Contemporanea, Tome 2 (2009) no. 2, pp. 207-215.

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For a transitive permutation group G on a finite set Ω, the Livingstone–Wagner Theorem states that if G is k-homogeneous then G is (k − 1)-transitive. It can be conjectured that the number of G-orbits on k-subsets of Ω is greater than or equal to the one on ordered (k − 1)-tuples of Ω, if |Ω| is sufficiently large. For the simplest case k = 3, we prove this by establishing a result on edge-colorings of complete digraphs.
DOI : 10.26493/1855-3974.92.46f
Keywords: Permutation group, Tranisitvity, Livingstone--Wagner Theorem 
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Yasuhiro Nakashima. A partial generalization of the Livingstone-Wagner Theorem. Ars Mathematica Contemporanea, Tome 2 (2009) no. 2, pp. 207-215. doi : 10.26493/1855-3974.92.46f. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.92.46f/

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