A partial generalization of the Livingstone-Wagner Theorem - Nakashima, Yasuhiro (Ars mathematica contemporanea, 2009)
For a transitive permutation group ?$G$? on a finite set ?$Ω$?, the Livingstone-Wagner Theorem states that if ?$G$? is ?$k$?-homogeneous and ?$2 \le k \le \frac{\vert Ω \vert}{2}$?, then ?$G$? is ?$(k - 1)$?-transitive. We conjecture that the number of ?$G$?-orbits on ?$k$?-subsets of ?$Ω$? is greater than or equal to the number of ?$G$?-orbits on ordered ?$(k - 1)$?-tuples of ?$Ω$?, if ?$\vert Ω \vert$? is sufficiently large. For the simplest case ?$k = 3$?, we verify this conjecture by establishing a result on edge-colorings of complete digraphs.
Keywords: grupe; matematika; orbite; teorija grup; tranzitivnost