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Question 1. Let $\kappa<\lambda$ are uncountable regular cardinal. For any coloring $c: \kappa\times \lambda \rightarrow \omega$, are there $A\in [\kappa]^\kappa$ and $B\in [\lambda]^\lambda$ such that $c(A\times B)$ is constant?

Question 2. If the answer is no. Is it possible add some condition to make the proposition hold?

PS. I think this kind of results should be know, if possible, can someone provide reference to me, I didn't read book about this topic.

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This is a large chapter of combinatorial set theory. Statements of this type are called polarized partition relations, and were studied by Erdos, Hajnal, Rado, and Shelah, among others. See e.g. the chapter of Hajnal and Larson in the Handbook of Set Theory.

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