Graphs with Linearly Bounded Ramsey Numbers

Graphs with Linearly Bounded Ramsey Numbers

On Size Ramsey Numbers of Graphs with Bounded Degree

Answering a question of J. Beck B2], we prove that there exists a graph G on n vertices with maximum degree three and the size Ramsey number ^ r(G) cn(log n) where and c are positive constants. For graphs G and F, write F ! G to mean that if the edges of F are colored by red and blue, then F contains a monochromatic copy of G. Erd} os, Faudree, Rousseau and Schelp EFRS] were the rst to consider...

On ordered Ramsey numbers of bounded-degree graphs

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 for every integer d ≥ 3, almost every d-regular graph G satisfies R(G) ≥ n3/2−1/d 4 log n log log n fo...

A transference principle for Ramsey numbers of bounded degree graphs

We investigate Ramsey numbers of bounded degree graphs and provide an interpolation between known results on the Ramsey numbers of general bounded degree graphs and bounded degree graphs of small bandwidth. Our main theorem implies that there exists a constant c such that for every ∆, there exists β such that if G is an n-vertex graph with maximum degree at most ∆ having a homomorphism f into a...

On graphs with linear Ramsey numbers

For a ®xed graph H, the Ramsey number r (H ) is de®ned to be the least integer N such that in any 2-coloring of the edges of the complete graph KN, some monochromatic copy of H is always formed. Let H(n, ) denote the class of graphs H having n vertices and maximum degree at most . It was shown by Chvata l, RoÈdl, SzemereÂdi, and Trotter that for each there exists c ( ) such that r (H )< c ( )n...