On two problems in graph Ramsey theory

Abstract

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every twocoloring of the edges of the complete graph KN contains a monochromatic copy of H. A famous result of Chv atal, Rodl, Szemer edi and Trotter states that there exists a constant c( Delta) such that r(H) c(Delta )n for every graph H with n vertices and maximum degree . The important open question is to determine the constant c(Delta ). The best results, both due to Graham, Rodl and Ruci nski, state that there are constants c and c0 such that 2c0 Delta[less than or equal to] c( Delta) 2c Deltalog2Delta . We improve this upper bound, showing that there is a constant c for which c(Delta )[less than or equal to] 2cDelta log Delta. The induced Ramsey number rind(H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erdos conjectured the existence of a constant c such that, for any graph H on n vertices, rind(H) 2cn. We move a step closer to proving this conjecture, showing that rind(H)[less than or equal to] 2cn log n. This improves upon an earlier result of Kohayakawa, Promel and Rodl by a factor of log n in the exponent.