156 research outputs found
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.
Flows and bisections in cubic graphs
A -weak bisection of a cubic graph is a partition of the vertex-set of
into two parts and of equal size, such that each connected
component of the subgraph of induced by () is a tree of at
most vertices. This notion can be viewed as a relaxed version of
nowhere-zero flows, as it directly follows from old results of Jaeger that
every cubic graph with a circular nowhere-zero -flow has a -weak bisection. In this paper we study problems related to the
existence of -weak bisections. We believe that every cubic graph which has a
perfect matching, other than the Petersen graph, admits a 4-weak bisection and
we present a family of cubic graphs with no perfect matching which do not admit
such a bisection. The main result of this article is that every cubic graph
admits a 5-weak bisection. When restricted to bridgeless graphs, that result
would be a consequence of the assertion of the 5-flow Conjecture and as such it
can be considered a (very small) step toward proving that assertion. However,
the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio
- …