Ramsey Numbers of Squares of Paths

The Ramsey number R(G,H) has been actively studied for the past 40 years, and it was determined for a large family of pairs (G,H) of graphs. The Ramsey number of paths was determined very early on, but surprisingly very little is known about the Ramsey number for the powers of paths. The r-th power of a path on n vertices is obtained by joining any two vertices with distance at most r.We show that for n large and discuss some related questions.