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 two-coloring of the edges of the complete graph KN contains a monochromatic copy of H. A famous result of Chv\'atal, R\"odl, Szemer\'edi 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, R\"odl and Ruci\'nski, state that there are constants c and c' such that 2c' ≤ c() ≤ 2c 2 . We improve this upper bound, showing that there is a constant c for which c() ≤ 2c . 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) ≤ 2c n. We move a step closer to proving this conjecture, showing that rind (H) ≤ 2c n n. This improves upon an earlier result of Kohayakawa, Pr\"omel and R\"odl by a factor of n in the exponent.