Replacing the host K_n by n-chromatic graphs in Ramsey-type results

Abstract

We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$ vertices. The second is the theorem of Cockayne and Lorimer stating that for positive integers satisfying $n_1=\max\{n_1,n_2,\dots,n_t\}$ and with $n=n_1+1+\sum_{i=1}^t (n_i-1)$, the following holds. In every coloring of the edges of $K_n$ with colors $1,2\dots,t$ there is a monochromatic matching of size $n_i$ for some $i\in \{1,2,\dots,t\}$.