356 research outputs found
Online Ramsey Numbers and the Subgraph Query Problem
The -online Ramsey game is a combinatorial game between two players,
Builder and Painter. Starting from an infinite set of isolated vertices,
Builder draws an edge on each turn and Painter immediately paints it red or
blue. Builder's goal is to force Painter to create either a red or a blue
using as few turns as possible. The online Ramsey number
is the minimum number of edges Builder needs to guarantee a win in the
-online Ramsey game. By analyzing the special case where Painter plays
randomly, we obtain an exponential improvement for the lower bound on the diagonal online Ramsey
number, as well as a corresponding improvement for the off-diagonal case, where is fixed
and . Using a different randomized Painter strategy, we
prove that , determining this function up
to a polylogarithmic factor. We also improve the upper bound in the
off-diagonal case for .
In connection with the online Ramsey game with a random Painter, we study the
problem of finding a copy of a target graph in a sufficiently large unknown
Erd\H{o}s--R\'{e}nyi random graph using as few queries as possible,
where each query reveals whether or not a particular pair of vertices are
adjacent. We call this problem the Subgraph Query Problem. We determine the
order of the number of queries needed for complete graphs up to five vertices
and prove general bounds for this problem.Comment: Corrected substantial error in the proof of Theorem
Towards Realistic String Vacua From Branes At Singularities
We report on progress towards constructing string models incorporating both
realistic D-brane matter content and moduli stabilisation with dynamical
low-scale supersymmetry breaking. The general framework is that of local
D-brane models embedded into the LARGE volume approach to moduli stabilisation.
We review quiver theories on del Pezzo () singularities including
both D3 and D7 branes. We provide supersymmetric examples with three
quark/lepton families and the gauge symmetries of the Standard, Left-Right
Symmetric, Pati-Salam and Trinification models, without unwanted chiral
exotics. We describe how the singularity structure leads to family symmetries
governing the Yukawa couplings which may give mass hierarchies among the
different generations. We outline how these models can be embedded into compact
Calabi-Yau compactifications with LARGE volume moduli stabilisation, and state
the minimal conditions for this to be possible. We study the general structure
of soft supersymmetry breaking. At the singularity all leading order
contributions to the soft terms (both gravity- and anomaly-mediation) vanish.
We enumerate subleading contributions and estimate their magnitude. We also
describe model-independent physical implications of this scenario. These
include the masses of anomalous and non-anomalous U(1)'s and the generic
existence of a new hyperweak force under which leptons and/or quarks could be
charged. We propose that such a gauge boson could be responsible for the ghost
muon anomaly recently found at the Tevatron's CDF detector.Comment: 40 pages, 10 figure
Hypergraph Ramsey numbers of cliques versus stars
Let denote the complete -uniform hypergraph on vertices
and the -uniform hypergraph on vertices consisting of all
edges incident to a given vertex. Whereas many hypergraph Ramsey
numbers grow either at most polynomially or at least exponentially, we show
that the off-diagonal Ramsey number exhibits an
unusual intermediate growth rate, namely, for some positive
constants and . The proof of these bounds brings in a novel Ramsey
problem on grid graphs which may be of independent interest: what is the
minimum such that any -edge-coloring of the Cartesian product contains either a red rectangle or a blue ?Comment: 13 page
Online Ramsey Numbers and the Subgraph Query Problem
The (m,n)-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red K_m or a blue K_n using as few turns as possible. The online Ramsey number [equation; see abstract in PDF for details] is the minimum number of edges Builder needs to guarantee a win in the (m,n)-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement
[equation; see abstract in PDF for details]
for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement
[equation; see abstract in PDF for details]
for the off-diagonal case, where m â„ 3 is fixed and n â â. Using a different randomized Painter strategy, we prove that [equation; see abstract in PDF for details], determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for m â„ 4.
In connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph H in a sufficiently large unknown ErdĆs-RĂ©nyi random graph G(N,p) using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem
Gravity waves and the LHC: Towards high-scale inflation with low-energy SUSY
It has been argued that rather generic features of string-inspired
inflationary theories with low-energy supersymmetry (SUSY) make it difficult to
achieve inflation with a Hubble scale H > m_{3/2}, where m_{3/2} is the
gravitino mass in the SUSY-breaking vacuum state. We present a class of
string-inspired supergravity realizations of chaotic inflation where a simple,
dynamical mechanism yields hierarchically small scales of post-inflationary
supersymmetry breaking. Within these toy models we can easily achieve small
ratios between m_{3/2} and the Hubble scale of inflation. This is possible
because the expectation value of the superpotential relaxes from large to
small values during the course of inflation. However, our toy models do not
provide a reasonable fit to cosmological data if one sets the SUSY-breaking
scale to m_{3/2} < TeV. Our work is a small step towards relieving the apparent
tension between high-scale inflation and low-scale supersymmetry breaking in
string compactifications.Comment: 21+1 pages, 5 figures, LaTeX, v2: added references, v3: very minor
changes, version to appear in JHE
M2-Branes and Fano 3-folds
A class of supersymmetric gauge theories arising from M2-branes probing
Calabi-Yau 4-folds which are cones over smooth toric Fano 3-folds is
investigated. For each model, the toric data of the mesonic moduli space is
derived using the forward algorithm. The generators of the mesonic moduli space
are determined using Hilbert series. The spectrum of scaling dimensions for
chiral operators is computed.Comment: 128 pages, 39 figures, 42 table
Single-Scale Natural SUSY
We consider the prospects for natural SUSY models consistent with current
data. Recent constraints make the standard paradigm unnatural so we consider
what could be a minimal extension consistent with what we now know. The most
promising such scenarios extend the MSSM with new tree-level Higgs interactions
that can lift its mass to at least 125 GeV and also allow for flavor-dependent
soft terms so that the third generation squarks are lighter than current bounds
on the first and second generation squarks. We argue that a common feature of
almost all such models is the need for a new scale near 10 TeV, such as a scale
of Higgsing or confinement of a new gauge group. We consider the question
whether such a model can naturally derive from a single mass scale associated
with supersymmetry breaking. Most such models simply postulate new scales,
leaving their proximity to the scale of MSSM soft terms a mystery. This
coincidence problem may be thought of as a mild tuning, analogous to the usual
mu problem. We find that a single mass scale origin is challenging, but suggest
that a more natural origin for such a new dynamical scale is the gravitino
mass, m_{3/2}, in theories where the MSSM soft terms are a loop factor below
m_{3/2}. As an example, we build a variant of the NMSSM where the singlet S is
composite, and the strong dynamics leading to compositeness is triggered by
masses of order m_{3/2} for some fields. Our focus is the Higgs sector, but our
model is compatible with a light stop (with the other generation squarks heavy,
or with R-parity violation or another mechanism to hide them from current
searches). All the interesting low-energy mass scales, including linear terms
for S playing a key role in EWSB, arise dynamically from the single scale
m_{3/2}. However, numerical coefficients from RG effects and wavefunction
factors in an extra dimension complicate the otherwise simple story.Comment: 32 pages, 3 figures; version accepted by JHE
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