513 research outputs found
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
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