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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
Intersecting families of discrete structures are typically trivial
The study of intersecting structures is central to extremal combinatorics. A
family of permutations is \emph{-intersecting} if
any two permutations in agree on some indices, and is
\emph{trivial} if all permutations in agree on the same
indices. A -uniform hypergraph is \emph{-intersecting} if any two of its
edges have vertices in common, and \emph{trivial} if all its edges share
the same vertices.
The fundamental problem is to determine how large an intersecting family can
be. Ellis, Friedgut and Pilpel proved that for sufficiently large with
respect to , the largest -intersecting families in are the trivial
ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest
-intersecting -uniform hypergraphs are also trivial when is large. We
determine the \emph{typical} structure of -intersecting families, extending
these results to show that almost all intersecting families are trivial. We
also obtain sparse analogues of these extremal results, showing that they hold
in random settings.
Our proofs use the Bollob\'as set-pairs inequality to bound the number of
maximal intersecting families, which can then be combined with known stability
theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira
result. Update 2: corrected statement of the unpublished Hamm--Kahn result,
and slightly modified notation in Theorem 1.6 Update 3: new title, updated
citations, and some minor correction
Triangle-Intersecting Families of Graphs
A family of graphs F is said to be triangle-intersecting if for any two
graphs G,H in F, the intersection of G and H contains a triangle. A conjecture
of Simonovits and Sos from 1976 states that the largest triangle-intersecting
families of graphs on a fixed set of n vertices are those obtained by fixing a
specific triangle and taking all graphs containing it, resulting in a family of
size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations
(for example, we prove that the same is true of odd-cycle-intersecting
families, and we obtain best possible bounds on the size of the family under
different, not necessarily uniform, measures). We also obtain stability
results, showing that almost-largest triangle-intersecting families have
approximately the same structure.Comment: 43 page
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