Addresses in a binary tree and MAD families with strong combinatorial properties

A classical problem of Erdős and Shelah (1972) asks whether there exists a MAD family whose elements cannot be “hidden” from any sufficiently large set. Such families are called completely separable, and it remains open whether they can be constructed in ZFC alone. In this talk we present the technique, originally due to Shelah and refined by Mildenberger–Raghavan–Steprāns, that yields a completely separable MAD family under the assumption 𝔰 ≤ 𝔞. The key idea is to assign to each member of the family an address in a binary tree: block-splitting families provide the splittings needed to grow a Cantor-scheme of positive sets, while incompatibility of addresses ensures almost disjointness at each stage of the recursion. We will give a detailed exposition of the proof and briefly discuss further applications of the method.