Sam Adam-Day (University of Oxford)

Rigid branchwise-real tree orders

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A branchwise-real tree order is a partial order tree in which every branch is isomorphic to a real interval. In this talk, I give several methods of constructing examples of these which are rigid (i.e. without non-trivial automorphisms), subject to increasing uniformity conditions. I show that there is a rigid branchwise-real tree order in which every branching point has the same degree, one in which every point is branching and of the same degree, and finally one in which every point is branching of the same degree and which admits no monotonic function into the reals. Trees are grown iteratively in stages, and a key technique is the construction (in ZFC) of a family of colourings of $(0,\infty)$ which is 'sufficiently generic', using these colourings to determine how to proceed with the construction.