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Aharoni and Korman (Order 9 (1992) 245--253) have conjectured that everyordered set without infinite antichains possesses a chain and a partition intoantichains so that each part intersects the chain. The conjecture is verifiedfor posets whose incomparability graph is locally finite. It follows that theconjecture is true for $(3 + 1)$-free posets with no infinite antichains.

The Aharoni--Korman conjecture for posets whose incomparability graph is locally finite