17,109 research outputs found
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
ON SYMMETRIC 3-WISE INTERSECTING FAMILIES
A family of sets is said to be symmetric if its automorphism group is
transitive, and -wise intersecting if any three sets in the family have
nonempty intersection. Frankl conjectured in 1981 that if is a
symmetric -wise intersecting family of subsets of , then
. Here, we give a short proof of Frankl's conjecture
using a 'sharp threshold' result of Friedgut and Kalai.Comment: 7 pages, typo corrected in description of 'tree' construction in
Section 3. Proc. Amer. Math. So
On symmetric intersecting families of vectors
A family of vectors is said to be intersecting if any two
elements of agree on at least one coordinate. We prove, for fixed , that the size of a symmetric intersecting subfamily of is ,
which is in stark contrast to the case of the Boolean hypercube (where ).
Our main contribution addresses limitations of existing technology: while there
is now some spectral machinery, developed by Ellis and the third author, to
tackle extremal problems in set theory involving symmetry, this machinery
relies crucially on the interplay between up-sets and biased product measures
on the Boolean hypercube, features that are notably absent in the problem at
hand; here, we describe a method for circumventing these barriers.Comment: 6 pages; It has been brought to our attention that our main result
(with slightly worse estimates) may be deduced from earlier work of Dinur,
Friedgut and Regev, and this revision acknowledges this fac
Cross-intersecting families and primitivity of symmetric systems
Let be a finite set and , the power set of ,
satisfying three conditions: (a) is an ideal in , that is,
if and , then ; (b) For with , if for any
with ; (c) for every . The
pair is called a symmetric system if there is a group
transitively acting on and preserving the ideal . A
family is said to be a
cross--family of if for any and with . We prove that if is a
symmetric system and is a
cross--family of , then where . This generalizes Hilton's theorem on
cross-intersecting families of finite sets, and provides analogs for
cross--intersecting families of finite sets, finite vector spaces and
permutations, etc.
Moreover, the primitivity of symmetric systems is introduced to characterize
the optimal families.Comment: 15 page
Conway groupoids, regular two-graphs and supersimple designs
A design is said to be supersimple
if distinct lines intersect in at most two points. From such a design, one can
construct a certain subset of Sym called a "Conway groupoid". The
construction generalizes Conway's construction of the groupoid . It
turns out that several infinite families of groupoids arise in this way, some
associated with 3-transposition groups, which have two additional properties.
Firstly the set of collinear point-triples forms a regular two-graph, and
secondly the symmetric difference of two intersecting lines is again a line. In
this paper, we show each of these properties corresponds to a group-theoretic
property on the groupoid and we classify the Conway groupoids and the
supersimple designs for which both of these two additional properties hold.Comment: 17 page
Regular Intersecting Families
We call a family of sets intersecting, if any two sets in the family
intersect. In this paper we investigate intersecting families of
-element subsets of such that every element of
lies in the same (or approximately the same) number of members of
. In particular, we show that we can guarantee if and only if .Comment: 15 pages, accepted versio
A factorization theorem for lozenge tilings of a hexagon with triangular holes
In this paper we present a combinatorial generalization of the fact that the
number of plane partitions that fit in a box is equal to
the number of such plane partitions that are symmetric, times the number of
such plane partitions for which the transpose is the same as the complement. We
use the equivalent phrasing of this identity in terms of symmetry classes of
lozenge tilings of a hexagon on the triangular lattice. Our generalization
consists of allowing the hexagon have certain symmetrically placed holes along
its horizontal symmetry axis. The special case when there are no holes can be
viewed as a new, simpler proof of the enumeration of symmetric plane
partitions.Comment: 20 page
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