1 research outputs found
On Mubayi's Conjecture and conditionally intersecting sets
Mubayi's Conjecture states that if is a family of -sized
subsets of which, for , satisfies whenever
for all distinct sets , then , with equality occurring only if is the family
of all -sized subsets containing some fixed element. This paper proves that
Mubayi's Conjecture is true for all families that are invariant with respect to
shifting; indeed, these families satisfy a stronger version of Mubayi's
Conjecture. Relevant to the conjecture, we prove a fundamental bijective
duality between -unstable families and -unstable families.
Generalising previous intersecting conditions, we introduce the
-conditionally intersecting condition for families of sets and prove
general results thereon. We conjecture on the size and extremal structures of
families that are -conditionally
intersecting but which are not intersecting, and prove results related to this
conjecture. We prove fundamental theorems on two -conditionally
intersecting families that generalise previous intersecting families, and we
pose an extension of a previous conjecture by Frankl and F\"uredi on
-conditionally intersecting families. Finally, we generalise a
classical result by Erd\H{o}s, Ko and Rado by proving tight upper bounds on the
size of -conditionally intersecting families and by characterising the families that attain these bounds. We extend
this theorem for certain parametres as well as for sufficiently large families
with respect to -conditionally intersecting families
whose members have at most a fixed number
members