Cardinal inequalities with Shanin number and \(\pi\)-character (Q6073806)

In this paper the authors present some new ZFC-consistent cardinal inequalities. In particular, they prove that, under GCH, \(|X| \leq sh(X)^{\pi\chi(X)\psi_c(X)}\) (hence \(|X| \leq sh(X)^{\chi(X)}\)) for any Hausdorff space \(X\) and, still under GCH, \(d(X) \leq sh(X)^ {t(X)\pi\chi(X)}\) for every regular space \(X\). Here \(sh(X)= \min\{\kappa \geq \omega: \kappa^+ \hbox{ is a caliber of } X\}\) denotes the Shanin number of \(X\). They also establish in ZFC that \(|X| \leq wL(X)^{\overline{\Delta}(X) 2^{\pi\chi(X)}}\) whenever \(X\) is a Urysohn space. Here \(\overline{\Delta}(X)\) denotes the regular diagonal degree of \(X\). Additionally, some important remarks and examples are given throughout the paper.