Ramsey numbers of ordered graphs

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.In contrast with the case of unordered graphs, 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. 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

Chromatic Number of Ordered Graphs with Forbidden Ordered Subgraphs

“…Ordered Ramsey numbers were recently studied by Conlon et al [7] and Balko et al [2]. Other research on ordered graphs includes characterizations of classes of graphs by forbidden ordered subgraphs [8,13] and the study of perfectly ordered graphs [6].…”

“…The third item in the following theorem is an immediate corollary of a result by Weidert [19] who provides a linear upper bound on the the extremal function for M . The other results are based on linear upper bounds for the extremal functions of nestings due to Dujmovic and Wood [10], on the extremal function of crossings due to Capoyleas and Pach [5] and lower bounds for ordered Ramsey numbers due to Conlon et al [7], see also Balko et al [2]. See Section 3 for a more detailed description of extremal functions and ordered Ramsey numbers.…”

“…• Conlon et al [7] and independently Balko et al [2] prove that that there is a positive constant c such that for any sufficiently large positive integer k there is an ordered matchings on k vertices with ordered Ramsey number at least 2 c log(k) 2 log log(k) . If, for some ordered graph H, the edges of a complete ordered graph G on N = R ≺ (H) − 1 vertices are colored in two colors without monochromatic copies of H, then both color classes form ordered graphs G 1 and G 2 in Forb ≺ (H).…”

Ramsey numbers and monotone colorings

“…(i) for every l ∈ L, there are at least k + 1 points of P lying on l, (ii) for every (k + 1)-tuple of distinct points of P , there is a unique curve l from L passing through each point of this (k + 1)-tuple, (iii) any two distinct curves from L cross at most k times. 1 This notion naturally generalizes the concept of generalized point sets [14] (sometimes called abstract order types), which correspond to 1-pseudoconfigurations. It also captures the essential combinatorial properties of configurations of points and graphs of polynomial functions, which is a setting considered by Eliáš and Matoušek [6] in their study of higher-order Erdős-Szekeres theorems.…”

Vertical Visibility Among Parallel Polygons in Three Dimensions

“…To prove Theorem 2 we used a Ramsey-type theorem [9, Theorem 2.1] for ordered graphs. We wonder if the recent developments in the Ramsey theory for ordered graphs [2,5] could shed more light on our problem.…”