9,066 research outputs found
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
All finite transitive graphs admit self-adjoint free semigroupoid algebras
In this paper we show that every non-cycle finite transitive directed graph
has a Cuntz-Krieger family whose WOT-closed algebra is . This
is accomplished through a new construction that reduces this problem to
in-degree -regular graphs, which is then treated by applying the periodic
Road Coloring Theorem of B\'eal and Perrin. As a consequence we show that
finite disjoint unions of finite transitive directed graphs are exactly those
finite graphs which admit self-adjoint free semigroupoid algebras.Comment: Added missing reference. 16 pages 2 figure
Two extensions of Ramsey's theorem
Ramsey's theorem, in the version of Erd\H{o}s and Szekeres, states that every
2-coloring of the edges of the complete graph on {1, 2,...,n} contains a
monochromatic clique of order 1/2\log n. In this paper, we consider two
well-studied extensions of Ramsey's theorem.
Improving a result of R\"odl, we show that there is a constant such
that every 2-coloring of the edges of the complete graph on \{2, 3,...,n\}
contains a monochromatic clique S for which the sum of 1/\log i over all
vertices i \in S is at least c\log\log\log n. This is tight up to the constant
factor c and answers a question of Erd\H{o}s from 1981.
Motivated by a problem in model theory, V\"a\"an\"anen asked whether for
every k there is an n such that the following holds. For every permutation \pi
of 1,...,k-1, every 2-coloring of the edges of the complete graph on {1, 2,
..., n} contains a monochromatic clique a_1<...<a_k with
a_{\pi(1)+1}-a_{\pi(1)}>a_{\pi(2)+1}-a_{\pi(2)}>...>a_{\pi(k-1)+1}-a_{\pi(k-1)}.
That is, not only do we want a monochromatic clique, but the differences
between consecutive vertices must satisfy a prescribed order. Alon and,
independently, Erd\H{o}s, Hajnal and Pach answered this question affirmatively.
Alon further conjectured that the true growth rate should be exponential in k.
We make progress towards this conjecture, obtaining an upper bound on n which
is exponential in a power of k. This improves a result of Shelah, who showed
that n is at most double-exponential in k.Comment: 21 pages, accepted for publication in Duke Math.
- …