7,462 research outputs found
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 Update 2: corrected statement of the unpublished Hamm--Kahn result,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
Erd\H{o}s-Ko-Rado for random hypergraphs: asymptotics and stability
We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for
the random -uniform hypergraph . For , let and . We show that with probability
tending to 1 as , the largest intersecting subhypergraph of
has size , for any . This lower bound on is
asymptotically best possible for . For this range of and ,
we are able to show stability as well.
A different behavior occurs when . In this case, the lower bound on
is almost optimal. Further, for the small interval , the largest intersecting subhypergraph of
has size , provided that .
Together with previous work of Balogh, Bohman and Mubayi, these results
settle the asymptotic size of the largest intersecting family in
, for essentially all values of and
Toward Realistic Intersecting D-Brane Models
We provide a pedagogical introduction to a recently studied class of
phenomenologically interesting string models, known as Intersecting D-Brane
Models. The gauge fields of the Standard-Model are localized on D-branes
wrapping certain compact cycles on an underlying geometry, whose intersections
can give rise to chiral fermions. We address the basic issues and also provide
an overview of the recent activity in this field. This article is intended to
serve non-experts with explanations of the fundamental aspects, and also to
provide some orientation for both experts and non-experts in this active field
of string phenomenology.Comment: 85 pages, 8 figures, Latex, Bibtex, v2: refs added, typos correcte
New supersymmetric vacua on solvmanifolds
We obtain new supersymmetric flux vacua of type II supergravities on
four-dimensional Minkowski times six-dimensional solvmanifolds. The orientifold
O4, O5, O6, O7, or O8-planes and D-branes are localized. All vacua are in
addition not T-dual to a vacuum on the torus. The corresponding solvmanifolds
are proven to be Calabi-Yau, with explicit metrics. Other Ricci flat
solvmanifolds are shown to be only K\"ahler.Comment: v2: few additions and minor modifications, published versio
D6-brane Splitting on Type IIA Orientifolds
We study the open-string moduli of supersymmetric D6-branes, addressing both
the string and field theory aspects of D6-brane splitting on Type IIA
orientifolds induced by open-string moduli Higgsing (i.e., their obtaining
VEVs). Specifically, we focus on the Z_2 x Z_2 orientifolds and address the
symmetry breaking pattern for D6-branes parallel with the orientifold 6-planes
as well as those positioned at angles. We demonstrate that the string theory
results, i.e., D6-brane splitting and relocating in internal space, are in one
to one correspondence with the field theory results associated with the
Higgsing of moduli in the antisymmetric representation of Sp(2N) gauge symmetry
(for branes parallel with orientifold planes) or adjoint representation of U(N)
(for branes at general angles). In particular, the moduli Higgsing in the
open-string sector results in the change of the gauge structure of D6-branes
and thus changes the chiral spectrum and family number as well. As a
by-product, we provide the new examples of the supersymmetric Standard-like
models with the electroweak sector arising from Sp(2N)_L x Sp(2N)_R gauge
symmetry; and one four-family example is free of chiral Standard Model exotics.Comment: 44 pages, 7 figures; The anomaly-free models in Subsections 4.2 and
4.3 presented, references added, typos fixe
Models of Particle Physics from Type IIB String Theory and F-theory: A Review
We review particle physics model building in type IIB string theory and
F-theory. This is a region in the landscape where in principle many of the key
ingredients required for a realistic model of particle physics can be combined
successfully. We begin by reviewing moduli stabilisation within this framework
and its implications for supersymmetry breaking. We then review model building
tools and developments in the weakly coupled type IIB limit, for both local
D3-branes at singularities and global models of intersecting D7-branes. Much of
recent model building work has been in the strongly coupled regime of F-theory
due to the presence of exceptional symmetries which allow for the construction
of phenomenologically appealing Grand Unified Theories. We review both local
and global F-theory model building starting from the fundamental concepts and
tools regarding how the gauge group, matter sector and operators arise, and
ranging to detailed phenomenological properties explored in the literature.Comment: 79 pages, Invited review article for the International Journal of
Modern Physics
Chiral D-brane Models with Frozen Open String Moduli
Most intersecting D-brane vacua in the literature contain additional massless
adjoint fields in their low energy spectrum. The existence of these additional
fields make it difficult to obtain negative beta functions and, eventually,
asymptotic freedom. We address this important issue for N=1 intersecting
D-brane models, rephrasing the problems in terms of (open string) moduli
stabilization. In particular, we consider a Z2 x Z2 orientifold construction
where D6-branes wrap rigid 3-cycles and such extra adjoint fields do not arise.
We derive the model building rules and consistency conditions for intersecting
branes in this background, and provide N=1 chiral vacua free of adjoint fields.
More precisely, we construct a Pati-Salam-like model whose SU(4) gauge group is
asymptotically free. We also comment on the application of these results for
obtaining gaugino condensation in chiral D-brane models. Finally, we embed our
constructions in the framework of flux compactification, and construct new
classes of N=1 and N=0 chiral flux vacua.Comment: 55 pages, 4 figures. Bibtex forma
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