Sat, 19 Aug 2023 10:25:46 +0000

Let's say I had a partition property $\kappa \to \omega_{\lambda}^n$ so that any coloring of sets of size $n$ with $\lambda$ many colors gives a homogeneous set of size $\omega$. Does anyone do any work that tries to guide the homogeneous set to be in a certain "place"? Design a coloring so that the homogeneous set cannot be in certain places? Just wondering if this is fun or useful.