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All -transitive groups satisfy the strict-EKR property
A subset of a transitive permutation group is
said to be an intersecting set if, for every , there is an such that . The stabilizer of a point in and
its cosets are intersecting sets of size . Such families are referred to
as canonical intersecting sets. A result by Meagher, Spiga, and Tiep states
that if is a -transitive group, then is the size of an
intersecting set of maximum size in . In some -transitive groups (for
instance , ), every intersecting set of
maximum possible size is canonical. A permutation group, in which every
intersecting family of maximum possible size is canonical, is said to satisfy
the strict-EKR property. In this article, we investigate the structure of
intersecting sets in -transitive groups. A conjecture by Meagher and Spiga
states that all -transitive groups satisfy the strict-EKR property. Meagher
and Spiga showed that this is true for the -transitive group
. Using the classification of -transitive groups and some
results in literature, the conjecture reduces to showing that the
-transitive group satisfies the strict-EKR property. We
show that satisfies the strict-EKR property and as a
consequence, we prove Meagher and Spiga's conjecture. We also prove a stronger
result for by showing that "large" intersecting sets in
must be a subset of a canonical intersecting set. This
phenomenon is called stability.Comment: 20 page