Graph Ramsey Theory for Multiple Copies of Graphs

Question

I had the following question from Graph Ramsey theory. Show that if $m \geq 2$, then $$ R((m+1)K _{3},K _{3})\geq R(mK _{3},K _{3}) + 3. $$

Thanks.

Answer 1

In

S. A. Burr, P. Erdõs, and J. H. Spencer. Ramsey theorems for multiple copies of graphs, Trans. Amer. Math. Soc., 209 (1975), 87-99. MR53 #13015,

it is shown that if $m\ge 2$ and $m\ge n\ge 1$, then $r(mK_3,nK_3)=3m+2n$. This gives your result immediately by taking $n=1$.