Let TsH be the graph obtained from a given graph H by subdividing each
edge s times. Motivated by a problem raised by Igor Pak [Mixing
time and long paths in graphs, in Proc. of the 13th annual ACM-SIAM
Symposium on Discrete Algorithms (SODA 2002) 321–328], we prove
that, for any graph H, there exist graphs G with O(s) edges that are
Ramsey with respect to TsH.