On the single-orbit conjecture for uncoverings-by-bases

Robert F. Bailey 1  and Peter J. Cameron 2
  • 1 Robert F. Bailey, School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada. E-mail: robertb@math.carleton.ca
  • 2 Peter J. Cameron, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom. E-mail: p.j.cameron@qmul.ac.uk


Let G be a permutation group acting on a finite set Ω. An uncovering-by-bases (or UBB) for G is a set of bases for G such that any r-subset of Ω is disjoint from at least one base in , where , for d the minimum degree of G. The single-orbit conjecture asserts that for any finite permutation group G, there exists a UBB for G contained in a single orbit of G on its irredundant bases. We prove a case of this conjecture, for when G is k-transitive and has a base of size k + 1. Furthermore, in the more restricted case when G is primitive and has a base of size 2, we show how to construct a UBB of minimum possible size.

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The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.