Uncountable-chromatic graphs have common \(4\)-chromatic subgraphs

A problem on 4-chromatic subgraphs (proposed by Erdös, Hajnal [1]):
Is it true that if \( G_1, G_2\) are \( \aleph_1\)-chromatic graphs then they have a common \( 4\)-chromatic subgraph?

Erdös, Hajnal and Shelah [2] proved that any \( \omega_1\)-chromatic graph contains all cycles \( C_k\) for \( k > k_0\). Consequently, the above problem has an affirmative answer for 3-chromatic graphs.


Bibliography
1 P. Erdös and A. Hajnal, Chromatic number of finite and infinite graphs and hypergraphs, Special volume on order sets and their applications (L'Arbresle 1982), Discrete Math. 53 (1985), 281-285.
2 P. Erdös, A. Hajnal and S. Shelah, On some general properties of chromatic numbers, Topics in topology (Proc. Colloq., Keszthely, 1972); Colloq. Math. Soc. János Bolyai, Vol. 8, 243-255, North-Holland, Amsterdam, 1974.