9,414 research outputs found
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
On symmetric intersecting families
We make some progress on a question of Babai from the 1970s, namely: for with , what is the largest possible cardinality
of an intersecting family of -element subsets of
admitting a transitive group of automorphisms? We give upper and lower bounds
for , and show in particular that as if and only if for some function
that increases without bound, thereby determining the threshold
at which `symmetric' intersecting families are negligibly small compared to the
maximum-sized intersecting families. We also exhibit connections to some basic
questions in group theory and additive number theory, and pose a number of
problems.Comment: Minor change to the statement (and proof) of Theorem 1.4; the authors
thank Nathan Keller and Omri Marcus for pointing out a mistake in the
previous versio
Minimum saturated families of sets
We call a family of subsets of -saturated if it
contains no pairwise disjoint sets, and moreover no set can be added to
while preserving this property (here ).
More than 40 years ago, Erd\H{o}s and Kleitman conjectured that an
-saturated family of subsets of has size at least . It is easy to show that every -saturated family has size at
least , but, as was mentioned by Frankl and Tokushige,
even obtaining a slightly better bound of , for some
fixed , seems difficult. In this note, we prove such a result,
showing that every -saturated family of subsets of has size at least
.
This lower bound is a consequence of a multipartite version of the problem,
in which we seek a lower bound on
where are families of subsets of ,
such that there are no pairwise disjoint sets, one from each family
, and furthermore no set can be added to any of the families
while preserving this property. We show that , which is tight e.g.\ by taking
to be empty, and letting the remaining families be the families
of all subsets of .Comment: 8 page
Invariance principle on the slice
We prove an invariance principle for functions on a slice of the Boolean
cube, which is the set of all vectors {0,1}^n with Hamming weight k. Our
invariance principle shows that a low-degree, low-influence function has
similar distributions on the slice, on the entire Boolean cube, and on Gaussian
space.
Our proof relies on a combination of ideas from analysis and probability,
algebra and combinatorics.
Our result imply a version of majority is stablest for functions on the
slice, a version of Bourgain's tail bound, and a version of the Kindler-Safra
theorem. As a corollary of the Kindler-Safra theorem, we prove a stability
result of Wilson's theorem for t-intersecting families of sets, improving on a
result of Friedgut.Comment: 36 page
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