Countably compact extensions and cardinal characteristics of the continuum

In this paper, we show that the existence of certain first-countable compact-like extensions is equivalent to the equality between corresponding cardinal characteristics of the continuum. For instance, $\mathfrak{b}=\mathfrak{s}=\mathfrak{c}$ if and only if every regular first-countable space of weight $<\mathfrak{c}$ can be densely embedded into a regular first-countable countably compact space.