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We present an elementary construction of an uncountably chromatic graph without uncountable, infinitely connected subgraphs., Comment: 3 pages; V2 corrects a mistake in the last line of the proof

We show that any graph that contains k edge-disjoint double rays for any k>0 contains also infinitely many edge-disjoint double rays. This was conjectured by Andreae in 1981., Comment: 15 pages, 2 figures

Thin sums matroids were introduced to extend the notion of representability to non-finitary matroids. We give a new criterion for testing when the thin sums construction gives a matroid. We show that thin sums matroids over thin families are precisely the duals of representable matroids (those arising from vector spaces). We also show that the class of tame thin sums matroids is closed under duality and under taking minors, by giving a new characterisation of the matroids in this class. Finally, we show that all the matroids naturally associated to an infinite graph are tame thin sums matroids., Comment: 24 pages