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We can't find such pair of bad order types. Indeed given $\gamma$ number of colors and $\theta$, $\psi$ we can find large saturated dense linear orders $\alpha, \beta$ such that for any $f: \alpha\times \beta \to \gamma$, there exists $X\subset \alpha, Y\subset \beta$ such that $tp(X)=tp(\alpha), tp(Y)=tp(\beta)$ such that $f''X\times Y$ is constant.