From Sierpinski-type colourings to Ulam-type matrices

Ulam matrices were introduced by Ulam in his study of the measure problem. Ulam’s construction applies to all successor cardinals Kappa, and later Hajnal extended the construction to apply to some limit cardinals as well. In my talk I will show how a colouring principle introduced by Sierpinski can be used to construct matrices with similar applications as the matrices of Ulam and Hajnal. I will also show how such colouring principles can be obtained from the existence of a non-trivial C-sequence on Kappa using walks on ordinals. As a consequence, the resulting matrices are more readily available than the matrices of Ulam and Hajnal.

The results I present are joint work with Assaf Rinot.