Embedding theorems for graphs establishing negative partition relations
We started to work on this paper with the following observation . If 4 is a graph on J~ 1 vertices and establishing the negative partition relation tt l ]) 2 then 4 is universal for countable graphs . This last statement means that every countable graph N is isomorphic to a spanned subgraph of q . We have stated a number of results and problems of the above type in our paper [4] written in 1971 . (See Problems VIII, IX, X on pp . 285-286 .) Quite a few of these problems will be solved or modified by the results of this paper . S. Shelah has made an important remark concerning our problem. He has shown that our starting result cannot be generalized for higher cardinals .