2 problems

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Aldous–Lyons conjecture on ends of uniform spanning trees
Aldous–Lyons conjecture. The UST of $(G,o)$ is one-ended if and only if $G$ is one-ended.
Conjecturesolvedends-of-graphs uniform-spanning-trees unimodular-random-graphsarXiv posted 2 days ago
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Finite Hamilton-cover conjecture for two-ended transitive graphs
Let $\Gamma$ be a two-ended transitive graph. A finite Hamilton cover is a finite collection of Hamiltonian spanning subgraphs whose union covers the relevant graph structure, as i…
Conjectureopenends-of-graphs hamilton-covers transitive-graphsarXiv posted 2 days ago

Aldous–Lyons conjecture on ends of uniform spanning trees