I gave an invited talk at the first Gdańsk Logic Conference, May 2023.
Talk Title: Was Ulam right?
Abstract: An Ulam matrix is one of the earliest gems of infinite combinatorics. Around the same time of its discovery, another Polish mathematician, Wacław Sierpiński was toying with pathological consequences of the continuum hypothesis. We shall argue that one of these consequences gives an improved Ulam matrix. We shall then embark into a systematic study of these principles, identifying a weakness in the definition of an Ulam matrix (and its extension due to Hajnal) in view of its intended application: preventing a kappa-complete proper ideal over an uncountable cardinal kappa from being weakly kappa-saturated. It will be shown that a cardinal kappa admits an Ulam-type matrix iff kappa admits a nontrivial C-sequence (a concept arising from the theory of walks on ordinals).
This is joint work with Tanmay Inamdar.
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