The Ramsey Number of Multiple Copies of a Graph

Abstract: Let H be a graph without isolated vertices. The Ramsey Number r(nH) is the minimum N such that every coloring of the edges of the complete graph on N vertices with red and blue contains n pairwise vertex-disjoint monochromatic copies of H of the same color.

In 1975, Burr, Erdős and Spencer established that r(nH) is a linear function of n for large enough n. In 1987, Burr proved a superexponential upper bound for when the long-term linear behavior starts.

In 2023, Bucic and Sudakov showed that the long-term linear behavior starts already when n is an exponential function of |H| using some of Burr’s techniques along with the absorption method.

In this seminar, we will discuss the proof of Bucic and Sudakov.

Posted on Nov 3, 2025 in Seminario de Grafos, Seminars