1,878 research outputs found
Constructing dense graphs with sublinear Hadwiger number
Mader asked to explicitly construct dense graphs for which the size of the
largest clique minor is sublinear in the number of vertices. Such graphs exist
as a random graph almost surely has this property. This question and variants
were popularized by Thomason over several articles. We answer these questions
by showing how to explicitly construct such graphs using blow-ups of small
graphs with this property. This leads to the study of a fractional variant of
the clique minor number, which may be of independent interest.Comment: 10 page
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
Induced Ramsey-type theorems
We present a unified approach to proving Ramsey-type theorems for graphs with
a forbidden induced subgraph which can be used to extend and improve the
earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham,
and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by
Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's
regularity lemma, thereby giving much better bounds. The same approach can be
also used to show that pseudo-random graphs have strong induced Ramsey
properties. This leads to explicit constructions for upper bounds on various
induced Ramsey numbers.Comment: 30 page
Density theorems for bipartite graphs and related Ramsey-type results
In this paper, we present several density-type theorems which show how to
find a copy of a sparse bipartite graph in a graph of positive density. Our
results imply several new bounds for classical problems in graph Ramsey theory
and improve and generalize earlier results of various researchers. The proofs
combine probabilistic arguments with some combinatorial ideas. In addition,
these techniques can be used to study properties of graphs with a forbidden
induced subgraph, edge intersection patterns in topological graphs, and to
obtain several other Ramsey-type statements
A short proof of the multidimensional Szemer\'edi theorem in the primes
Tao conjectured that every dense subset of , the -tuples of
primes, contains constellations of any given shape. This was very recently
proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler.
Here we give a simple proof using the Green-Tao theorem on linear equations in
primes and the Furstenberg-Katznelson multidimensional Szemer\'edi theorem.Comment: 5 page
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