13,721 research outputs found
Lines in Euclidean Ramsey theory
Let be a sequence of points on a line with consecutive points of
distance one. For every natural number , we prove the existence of a
red/blue-coloring of containing no red copy of and no
blue copy of for any . This is best possible up to the
constant in the exponent. It also answers a question of Erd\H{o}s, Graham,
Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every
natural number , there is a set and a
red/blue-coloring of containing no red copy of and no
blue copy of .Comment: 7 page
Graph removal lemmas
The graph removal lemma states that any graph on n vertices with o(n^{v(H)})
copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite
its innocent appearance, this lemma and its extensions have several important
consequences in number theory, discrete geometry, graph theory and computer
science. In this survey we discuss these lemmas, focusing in particular on
recent improvements to their quantitative aspects.Comment: 35 page
Do the Electrons and Ions in X-ray Clusters Share the Same Temperature?
The virialization shock around an X-ray cluster primarily heats the ions,
since they carry most of the kinetic energy of the infalling gas. Subsequently,
the ions share their thermal energy with the electrons through Coulomb
collisions. We quantify the expected temperature difference between the
electrons and ions as a function of radius and time, based on a spherical
self-similar model for the accretion of gas by a cluster in an Omega=1, h=0.5
universe. Clusters with X-ray temperatures T=(4-10)*10^7 K, show noticeable
differences between their electron and ion temperatures at radii >2 Mpc. High
resolution spectroscopy with future X-ray satellites such as Astro E may be
able to determine the ion temperature in intracluster gas from the width of its
X-ray emission lines, and compare it to the electron temperature as inferred
from the free-free emission spectrum. Any difference between these temperatures
can be used to date the period of time that has passed since the infalling gas
joined the cluster.Comment: 18 pages, 3 figures, submitted to Ap
Exotic Decays of Heavy B quarks
Heavy vector-like quarks of charge , , have been searched for at the
LHC through the decays . In models where the
quark also carries charge under a new gauge group, new decay channels may
dominate. We focus on the case where the is charged under a
and describe simple models where the dominant decay mode is . With the inclusion of dark matter such
models can explain the excess of gamma rays from the Galactic center. We
develop a search strategy for this decay chain and estimate that with
integrated luminosity of 300 fb the LHC will have the potential to
discover both the and the for quarks with mass below
TeV, for a broad range of masses. A high-luminosity run can extend this
reach to TeV.Comment: 28 pages, 10 figures, 3 table
Large subgraphs without complete bipartite graphs
In this note, we answer the following question of Foucaud, Krivelevich and
Perarnau. What is the size of the largest -free subgraph one can
guarantee in every graph with edges? We also discuss the analogous
problem for hypergraphs.Comment: 4 page
Hedgehogs are not colour blind
We exhibit a family of -uniform hypergraphs with the property that their
-colour Ramsey numbers grow polynomially in the number of vertices, while
their -colour Ramsey numbers grow exponentially. This is the first example
of a class of hypergraphs whose Ramsey numbers show a strong dependence on the
number of colours.Comment: 7 page
Books versus triangles at the extremal density
A celebrated result of Mantel shows that every graph on vertices with
edges must contain a triangle. A robust version of
this result, due to Rademacher, says that there must in fact be at least
triangles in any such graph. Another strengthening, due
to the combined efforts of many authors starting with Erd\H{o}s, says that any
such graph must have an edge which is contained in at least triangles.
Following Mubayi, we study the interplay between these two results, that is,
between the number of triangles in such graphs and their book number, the
largest number of triangles sharing an edge. Among other results, Mubayi showed
that for any such that any graph
on vertices with at least edges and book number
at most contains at least triangles. He also
asked for a more precise estimate for in terms of . We make a
conjecture about this dependency and prove this conjecture for
and for , thereby answering Mubayi's question in these
ranges.Comment: 15 page
Cycle packing
In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every
graph on n vertices can be partitioned into O(n) cycles and edges. They
observed that one can easily get an O(n log n) upper bound by repeatedly
removing the edges of the longest cycle. We make the first progress on this
problem, showing that O(n log log n) cycles and edges suffice. We also prove
the Erd\H{o}s-Gallai conjecture for random graphs and for graphs with linear
minimum degree.Comment: 18 page
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