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Published in last 50 years
The following question is answered: If the real line is partitioned into countable sets, is there a Hamel basis that picks at most one element from each member of the partition?
This paper contains a proof of a conjecture of S. Simpson which is an infinitary version of a conjecture of Rota concerning partitions of finite dimensional vector spaces over a finite field. Rota's conjecture was proved by Graham, Leeb, and Rothschild ( Adv. in Math. 8 (1972), 417–433).
On partitions of finite vector spaces of small dimensions
Partitions of finite vector spaces: An application of the frobenius number in geometry