A note on the Size-Ramsey number of long subdivisions of graphs

Donadelli, Jair; Haxell, Penny E.; Kohayakawa, Yoshiharu

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 191-206.

Let $T_{s}H$ 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 $O\left(s\right)$ edges that are Ramsey with respect to $T_{s}H$.

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