On two problems in graph Ramsey theory

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 Chvátal, Rödl, Szemerédi and Trotter states that there exists a constant c(∆) such that r(H) ≤ c(∆)n for every graph H with n vertices and maximum degree ∆. The important open question is to determine the constant c(∆). The best results, both due to Graham, Rödl and Ruciński, state that there are constants c and c′ such that 2 ′∆ ≤ c(∆) ≤ 2 log . We improve this upper bound, showing that there is a constant c for which c(∆) ≤ 2 log . 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. Erdős conjectured the existence of a constant c such that, for any graph H on n vertices, rind(H) ≤ 2. We move a step closer to proving this conjecture, showing that rind(H) ≤ 2 log . This improves upon an earlier result of Kohayakawa, Prömel and Rödl by a factor of log n in the exponent.