We study connections between distributed local algorithms, finitary factors of iid processes,
and descriptive combinatorics in the context of regular trees. We extend the Borel …

In this article we survey the emerging field of descriptive graph combinatorics. This area has
developed in the last two decades or so at the interface of descriptive set theory and graph …

For an r-tuple. 1;:::; r/of special orthogonal dd matrices, we say that the Euclidean. d 1/-
dimensional sphere Sd1 is. 1;:::; r/-divisible if there is a subset A Â Sd1 such that its …

For an $ r $-tuple $(\gamma_1,\ldots,\gamma_r) $ of special orthogonal $ d\times d $
matrices, we say that the Euclidean $(d-1) $-dimensional sphere $ S^{d-1} $ is …

We show that every $ d $-regular bipartite Borel graph admits a Baire measurable $ k $-
regular spanning subgraph if and only if $ d $ is odd or $ k $ is even. This gives the first …

This thesis is about finding solutions to combinatorial problems on graphs such as proper
coloring or perfect matching in the presence of extra definability constraints. For example …

Let $\Gamma=\prod_ {i\in\mathbb {N}}\Gamma_i $ be an infinite product of nontrivial finite
groups, or let $\Gamma=(\mathbb {R}/\mathbb {Z})^ n $ be a finite-dimensional torus. For a …