644 research outputs found
On ordered Ramsey numbers of matchings versus triangles
For graphs and with linearly ordered vertex sets, the \ordered
Ramsey number is the smallest positive integer such that any
red-blue coloring of the edges of the complete ordered graph on
vertices contains either a blue copy of or a red copy of . Motivated
by a problem of Conlon, Fox, Lee, and Sudakov (2017), we study the numbers
where is an ordered matching on vertices.
We prove that almost all -vertex ordered matchings with interval
chromatic number 2 satisfy and
, improving a recent result by Rohatgi (2019).
We also show that there are -vertex ordered matchings with interval
chromatic number at least 3 satisfying , which asymptotically matches the best known lower bound on these
off-diagonal ordered Ramsey numbers for general -vertex ordered matchings.Comment: 16 pages, 2 figures; extended abstract to appear at EuroComb 202
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
Ordered Ramsey numbers
Given a labeled graph with vertex set , the ordered
Ramsey number is the minimum such that every two-coloring of the
edges of the complete graph on contains a copy of with
vertices appearing in the same order as in . The ordered Ramsey number of a
labeled graph is at least the Ramsey number and the two coincide for
complete graphs. However, we prove that even for matchings there are labelings
where the ordered Ramsey number is superpolynomial in the number of vertices.
Among other results, we also prove a general upper bound on ordered Ramsey
numbers which implies that there exists a constant such that for any labeled graph on vertex set .Comment: 27 page
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