38,114 research outputs found
Ramsey numbers for partially-ordered sets
We present a refinement of Ramsey numbers by considering graphs with a
partial ordering on their vertices. This is a natural extension of the ordered
Ramsey numbers. We formalize situations in which we can use arbitrary families
of partially-ordered sets to form host graphs for Ramsey problems. We explore
connections to well studied Tur\'an-type problems in partially-ordered sets,
particularly those in the Boolean lattice. We find a strong difference between
Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the
partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl
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 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
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
On the Geometric Ramsey Number of Outerplanar Graphs
We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2
outerplanar triangulations in both convex and general cases. We also prove that
the geometric Ramsey numbers of the ladder graph on vertices are bounded
by and , in the convex and general case, respectively. We
then apply similar methods to prove an upper bound on the
Ramsey number of a path with ordered vertices.Comment: 15 pages, 7 figure
Edge-ordered Ramsey numbers
We introduce and study a variant of Ramsey numbers for edge-ordered graphs,
that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey
number of an edge-ordered graph
is the minimum positive integer such that there exists an edge-ordered
complete graph on vertices such that every 2-coloring of
the edges of contains a monochromatic copy of
as an edge-ordered subgraph of .
We prove that the edge-ordered Ramsey number
is finite for every edge-ordered graph and we obtain better
estimates for special classes of edge-ordered graphs. In particular, we prove
for every bipartite
edge-ordered graph on vertices. We also introduce a natural
class of edge-orderings, called lexicographic edge-orderings, for which we can
prove much better upper bounds on the corresponding edge-ordered Ramsey
numbers.Comment: Minor revision, 16 pages, 1 figure. An extended abstract of this
paper will appeared in the Eurocomb 2019 proceedings in Acta Mathematica
Universitatis Comenianae. The paper has been accepted to the European Journal
of Combinatoric
EDGE-ORDERED RAMSEY NUMBERS
We introduce and study a variant of Ramsey numbers for edge-ordered graphs, that is, graphs with linearly ordered sets of edges.
The edge-ordered Ramsey number R_e(G) of an edge-ordered graph G is the minimum positive integer N such that there exists an edge-ordered complete graph K_N on N vertices such that every 2-coloring of the edges of K_N contains a monochromatic copy of G as an edge-ordered subgraph of K_N.
We prove that the edge-ordered Ramsey number R_e(G) is finite for every edge-ordered graph G and we obtain better estimates for special classes of edge-ordered graphs.
In particular, we prove R_e(G) <= 2^{O(n^3\log{n})} for every bipartite edge-ordered graph G on n vertices.
We also introduce a natural class of edge-orderings, called \emph{lexicographic edge-orderings}, for which we can prove much better upper bounds on the corresponding edge-ordered Ramsey numbers
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