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Ordered Ramsey numbers of loose paths and matchings
For a -uniform hypergraph with vertex set , the
ordered Ramsey number is the least integer such
that every -coloring of the edges of the complete -uniform graph on
vertex set contains a monochromatic copy of whose vertices
follow the prescribed order. Due to this added order restriction, the ordered
Ramsey numbers can be much larger than the usual graph Ramsey numbers. We
determine that the ordered Ramsey numbers of loose paths under a monotone order
grows as a tower of height one less than the maximum degree. We also extend
theorems of Conlon, Fox, Lee, and Sudakov [Ordered Ramsey numbers,
arXiv:1410.5292] on the ordered Ramsey numbers of 2-uniform matchings to
provide upper bounds on the ordered Ramsey number of -uniform matchings
under certain orderings.Comment: 13 page
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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