Abstract
An ordered graph is a pair G=(G,<) where G is a graph and < is a total ordering of its vertices. The ordered Ramsey number R(G) is the minimum number N such that every ordered complete graph with N vertices and with edges colored by two colors contains a monochromatic copy of G.
We show that there are arbitrarily large ordered matchings M_n on n vertices for which R(M_n) is superpolynomial in n. 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 R(G) is polynomial in the number of vertices of G if the bandwidth of G is constant or if G is an ordered graph of constant degeneracy and constant interval chromatic number.