SOME PROBLEMS ON PINITR AND INFINITE GRAPHS

Many of the problems Hajnal and I posed 15 years ago have been solved positively or negatively or shown to be undecidable. I state soae of the remaining ones and add a few new ones. I do not give a complete list of references but add a few where it seem essential. 1. Determine the a's for which oa + (oa,3)3. a must be a power of 0. I offer $1000 for a complete characterization and $250 for 2 a=w, the first open case. A short proof of a partition theorem for the ordinal ww. Ann. Math. Logic 6(1973/74). 2. Is it true that if a + (a,3): then also o + (a ,n)I? 3. c + (w+n,rl)i is an old result of Rado and myself. o+n can perhaps be replaced by any Q < w1 and4byany n<o,butIknow nothing about this. Hajnal and I proved UT + (oI.~,3)~. To our annoyance we could never show 0; + (w,o,4);. During this meeting Baumgartner and Hajnal proved, assuming the continuum hypothesis Primary Subject Classification D4A20.