Cichon's diagram describes the relationships between 12 infinite cardinal numbers, among them

* \(\aleph_1\) and \(\mathfrak{c}\)=continuum (the smallest and largest of the 12)

* \(\operatorname{cov(N)}\) and \(\operatorname{cov(M)}\), the covering numbers of the ideals of Lebesgue null and meager sets, respectively i.e., the answers to the questions "how many null/meager sets do we need to cover the reals?"

* \(\mathfrak{b}\) and \(\mathfrak{d}\), the bounding and dominating numbers. (The dominating number \(\mathfrak{d}\) is the smallest size of a family of \(\sigma\)-compact sets covering the irrationals.)

"ZFC + \(\mathfrak{c}=\aleph_1\)" implies of course that all 12 cardinals have the same value. All possible consequences for Cichon's diagram of the axioms "ZFC + \(\mathfrak{c}=\aleph_2\)" are known, or in other words: for every assignment of the values \(\aleph_1\) and \(\aleph_2\) to the cardinals in Cichon's diagram it is known whether there is a ZFC-universe in which this assignment is realized.

It is notoriously difficult to construct universes with prescribed properties in which \(\mathfrak{c}\) is large (even just larger than \(\aleph_2\)).

After discussing background and known results, I will highlight some features of a construction that will appear in a joint paper with Arthur Fischer, Jakob Kellner and Saharon Shelah. Our construction is a "creature iteration" (which is almost, but not quite, entirely unlike a product of creature forcings). All universes we construct will satisfy \(\mathfrak{d}=\aleph_1\) (which implies, among others, \(\operatorname{cov(M)}=\aleph_1\)), while the cardinals that are not obviously bounded by \(\mathfrak{d}\) can have (almost) arbitrary regular values.