Exploring the Limits of Independence in Mathematical Theories

Ever wondered how mathematical theories can be organized based on their complexity? Let's dive into the world of model theory, where we're looking at something called the $\mathrm{NSOP}_{2^{n+1}+1}$ hierarchy. This hierarchy is all about understanding the different levels of independence in formulas. Imagine you have a formula that divides over a model. If it does this for every sequence that's independent in a certain way, we say it $(n+1)$-$\eth$-divides. And if it implies a bunch of formulas that $(n+1)$-$\eth$-divide, it's called $(n+1)$-$\eth$-forking.Now, here's where it gets interesting. If a theory has $n$-$\eth$-independence that's either symmetric or transitive, then that theory is $\mathrm{NSOP}_{2^{n+1}+1}$. But the big question is: does symmetry or transitivity of $n$-$\eth$-independence always mean a theory is $\mathrm{NSOP}_{2^{n+1}+1}$? The jury's still out on that one.