Similar problems were investigated by Erd\H{o}s, Galvin, and Hajnal \cite{EGH75}. Erd\H{o}s claims that for graphs the problem is completely solved: a graph of chromatic number $\geq \aleph_1$ must contain all finite bipartite graphs but need not contain any fixed odd cycle.


References


[EGH75] Erd\H{o}s, P. and Galvin, F. and Hajnal, A., On set-systems having large chromatic number and not containing prescribed subsystems. (1975), 425--513.