\input zb-basic \input zb-ioport \iteman{io-port 00887764} \itemau{Hajnal, A.} \itemti{True embedding partition relations.} \itemso{Sauer, N. W. (ed.) et al., Finite and infinite combinatorics in sets and logic. Proceedings of the NATO Advanced Study Institute, Banff, Canada, April 21-May 4, 1991. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 411, 135-151 (1993).} \itemab Summary: This paper is primarily a survey of recent results of the author and others on transfinite generalizations of Ne\v{s}et\v{r}il-R\"odl type Ramsey theory. In the second half of the paper large universal graphs are studied in detail. \itemrv{~} \itemcc{} \itemut{partition relations; survey; Ramsey theory; large universal graphs} \itemli{} \end