5,351 research outputs found
Size-Ramsey numbers of structurally sparse graphs
Size-Ramsey numbers are a central notion in combinatorics and have been
widely studied since their introduction by Erd\H{o}s, Faudree, Rousseau and
Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of
-vertex graphs with constant maximum degree . For example, graphs
which also have constant treewidth are known to have linear size-Ramsey
numbers. On the other extreme, the canonical examples of graphs of unbounded
treewidth are the grid graphs, for which the best known bound has only very
recently been improved from to by Conlon, Nenadov and
Truji\'c. In this paper, we prove a common generalization of these results by
establishing new bounds on the size-Ramsey numbers in terms of treewidth (which
may grow as a function of ). As a special case, this yields a bound of
for proper minor-closed classes of graphs. In
particular, this bound applies to planar graphs, addressing a question of Wood.
Our proof combines methods from structural graph theory and classic
Ramsey-theoretic embedding techniques, taking advantage of the product
structure exhibited by graphs with bounded treewidth.Comment: 21 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
On-line Ramsey numbers
Consider the following game between two players, Builder and Painter. Builder
draws edges one at a time and Painter colours them, in either red or blue, as
each appears. Builder's aim is to force Painter to draw a monochromatic copy of
a fixed graph G. The minimum number of edges which Builder must draw,
regardless of Painter's strategy, in order to guarantee that this happens is
known as the on-line Ramsey number \tilde{r}(G) of G. Our main result, relating
to the conjecture that \tilde{r}(K_t) = o(\binom{r(t)}{2}), is that there
exists a constant c > 1 such that \tilde{r}(K_t) \leq c^{-t} \binom{r(t)}{2}
for infinitely many values of t. We also prove a more specific upper bound for
this number, showing that there exists a constant c such that \tilde{r}(K_t)
\leq t^{-c \frac{\log t}{\log \log t}} 4^t. Finally, we prove a new upper bound
for the on-line Ramsey number of the complete bipartite graph K_{t,t}.Comment: 11 page
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