If $\lambda<\kappa$ are two regular cardinals, then every locally $\lambda$-presentable category is locally $\kappa$-presentable. The analogous statement does not hold for $\lambda$-accessible categories, though. In the books by Adamek-Rosicky and Makkai-Paré a special relation $\lambda\vartriangleleft \kappa$ for this property is introduced. It holds if and only if for every set $X$ with $\# X < \kappa$ the partial order $P_{<\lambda}(X)$ of subsets of $X$ with $<\lambda$ elements has a cofinal subset of cardinality $<\kappa$.
Several properties are established (see also MO/150305) which help to study accessible categories. But in principle this relation between regular cardinals is a purely set theoretic concept. Therefore, I wonder if set theorists have actually considered to use this relation, and which applications outside of category theory and categorical logic are there.
