  Galeotti, Khomskii and V\"a\"an\"aanen recently introduced the notion of the
upward L\"owenheim-Skolem-Tarski number for a logic, strengthening the
classical notion of a Hanf number. A cardinal $\kappa$ is the \emph{upward
L\"owenheim-Skolem-Tarski number} (ULST) of a logic $\mathcal L$ if it is the
least cardinal with the property that whenever $M$ is a model of size at least
$\kappa$ satisfying a sentence $\varphi$ in $\mathcal L$, then there are
arbitrarily large models satisfying $\varphi$ and having $M$ as a substructure.
The substructure requirement is what differentiates the ULST number from the
Hanf number and gives the notion large cardinal strength. While it is a theorem
of ZFC that every logic has a Hanf number, Galeotti, Khomskii and
V\"a\"an\"anen showed that the existence of the ULST number for second-order
logic implies the existence of a partially extendible cardinal. We answer
positively their conjecture that the ULST number for second-order logic is the
least extendible cardinal.
  We define the strong ULST number by strengthening the substructure
requirement to elementary substructure. We investigate the ULST and strong ULST
numbers for several classical strong logics: infinitary logics, the
equicardinality logic, logic with the well-foundedness quantifier, second-order
logic, and sort logics. We show that the ULST and the strong ULST numbers are
characterized in some cases by classical large cardinals and in some cases by
natural new large cardinal notions that they give rise to. We show that for
some logics the notions of the ULST number, strong ULST number and least strong
compactness cardinal coincide, while for others, it is consistent that they can
be separated. Finally, we introduce a natural large cardinal notion
characterizing strong compactness cardinals for the equicardinality logic.
