On ordered Ramsey numbers of bounded-degree 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 2-coloring of the edges of the ordered complete graph on N vertices contains a monochromatic copy of G.We show that for every integer d ≥ 3, almost every d-regular graph G satisfies R(G) ≥ n 3/2−1/d 4 log n log log n for every ordering G of G. In particular, there are 3-regular graphs G on n vertices for which the numbers R(G) are superlinear in n, regardless of the ordering G of G. This solves a problem of Conlon, Fox, Lee, and Sudakov. On the other hand, we prove that every graph G on n vertices with maximum degree 2 admits an ordering G of G such that R(G) is linear in n.We also show that almost every ordered matching M with n vertices and with interval chromatic number two satisfies R(M) ≥ cn 2 / log 2 n for some absolute constant c. arXiv:1606.05628v2 [math.CO]

Ramsey numbers of ordered graphs under graph operations

“…Ordered Ramsey theory has become increasingly popular in recent years. In a series of papers Balko et al [1,2], and independently Conlon et al [3] investigated connections between some natural ordered graph parameters and the corresponding Ramsey numbers, as well as some striking differences between Ramsey numbers of classical and ordered sparse graphs.…”

“…However, the link between the interval chromatic number of a given ordered graph G and its Ramsey number is not clear. Indeed, there exist interval 2-chromatic orderings M n of the matching on n vertices, whose Ramsey number is of the order n 2−o (1) [3,2], while it is well-known that the Ramsey number R(M n ) of a matching on n vertices is linear in n. In fact, some ordered matchings (independent of their interval-chromatic number) have much higher Ramsey number: 1 The degeneracy of an ordered graph G is the degeneracy of the corresponding unordered graph; that is, the smallest integer d such that there exists an ordering of its vertices in which each vertex v has at most d neighbours w with w < v in the ordering.…”

“…has Ramsey number at most 2n + 1 by a simple pigeonhole argument. This prompts the question of minimizing the Ramsey number over all orderings of a given simple graph G. There has been some recent progress on this question by Balko, Jelínek and Valtr [2] who showed that every graph G on n vertices with maximum degree 2 admits an ordering whose Ramsey number is linear in n.…”

On Ordered Ramsey Numbers of Tripartite 3-Uniform Hypergraphs

“…We note that even for K (3) n the best known lower bound on R(K (3) n ) is of order 2 Ω(n 2 ) ; see (1).…”

“…The Ramsey numbers R(K (k) n ) are even less understood for k ≥ 3. For example, it is only known that 2 Ω(n 2 ) ≤ R(K (3) n ) ≤ 2 2 O(n) , (…”