I wonder if there are regular cardinals $\lambda$ and $\kappa$ such that $\kappa < \lambda \leq 2^\kappa$ and such that, consistently, the following holds:


Let $c: [\lambda]^2 \to \kappa$ be such that for $\alpha < \beta < \gamma$, we have $c(\alpha, \gamma) \leq \max \lbrace c(\alpha, \beta) , c(\beta, \gamma) \rbrace$. The there is a cofinal $H \subseteq \lambda$ such that $range(c \restriction [H]^2) \subseteq \eta$ for some $\eta < \kappa$.


Now, the second part is obviously true, even without the max condition, in case $\lambda $ is weakly compact, but then we lose $\lambda \leq 2^\kappa$ since $\lambda$ is strong limit.


I am particularly interested in the case where $\kappa = \aleph\_0$, or more generally a regular cardinal.



Ramsey-like property with order condition
A coloring $c:[\kappa]^2\rightarrow\theta$ is *subadditive of the first kind* if for all $\alpha<\beta<\gamma<\kappa$, $c(\alpha,\gamma)\le\max\{c(\alpha,\beta),c(\beta,\gamma)\}$. It is *subadditive of the second kind* if for all $\alpha<\beta<\gamma<\kappa$, $c(\alpha,\beta)\le\max\{c(\alpha,\gamma),c(\beta,\gamma)\}$. The combination of both is called *subadditive*, and is the subject matter of the following paper: <http://www.assafrinot.com/paper/36>


The Ramsey-theoretic results you are after may be found as Lemma 3.38(2), Corollary 3.42(2), and Theorem 3.45(2) of the above paper. In particular, assuming PFA, for every regular cardinal $\kappa>2^{\aleph\_0}$ that is not the successor of a singular cardinal of countable cofinality, every coloring $c:[\kappa]^2\rightarrow\omega$ that is subadditive of the first kind admits a cofinal subset $A\subseteq\kappa$ such that $c``[A]^2$ is finite.


