We discuss some well-known compactness principles for uncountable structures of small regular sizes \(\omega_n\) for \(2 \le n<\omega\), \(\aleph_{\omega+1}\), \(\aleph_{\omega^2+1}\), etc.), consistent from weakly compact (the size-restricted versions) or strongly compact or supercompact cardinals (the unrestricted versions). For the exposition, we divide the principles into logical principles, which are related to cofinal branches in trees and more general structures (various tree properties), and mathematical principles, which directly postulate compactness for structures like groups, graphs, or topological spaces (for instance, countable chromatic and color compactness of graphs, compactness of abelian groups, \(\Delta\)-reflection, Fodor-type reflection principle, and Rado's Conjecture).

We also focus on indestructibility, or preservation, of these principles in forcing extensions. While preservation adds a degree of robustness to such principles, it also limits their provable consequences. For example, some well-known mathematical problems such as Suslin Hypothesis, Whitehead's Conjecture, Kaplansky's Conjecture, and the categoricity of \(\omega_1\)-dense subsets of the reals (Baumgartner's Axiom), are independent from some of the strongest forms of compactness at \(\omega_2\). This is a refined version of Solovay's theorem that large cardinals are preserved by small forcings and hence cannot decide many natural problems in mathematics. Additionally, we observe that Rado's Conjecture plus \(2^\omega = \omega_2\) is consistent with the negative solutions, i.e. as they hold in \(V =L\), of some of these conjectures (Suslin's, Whitehead's, and Baumgartner's axiom), verifying that they hold in suitable Mitchell models.

Finally, we comment on whether the compactness principles under discussion are good candidates for axioms. We consider their consequences and the existence or non-existence of convincing unifications (such as Martin's Maximum or Rado's Conjecture). This part is a modest follow-up to the articles by Foreman “Generic large cardinals: new axioms for mathematics?” and Feferman et al. "Does mathematics need new axioms?''.