Abstract
There is a product of two linear orders of size \(2^{\aleph _0 } \) with the property that every subset or complement thereof contains a maximal chain. Furthermore, for regular ℵα, there is a product of two linear orders of size ℵα+2 that when colored with fewer than ℵα colors always has a monochromatic maximal chain. As a corollary, for every uncountable strong limit cardinal κ, there is an ordered set of cardinality κ that must be colored with at least κ colors before no monochromatic maximal chains are present. Duals of these results show that at least as much is true for maximal antichains.
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Communicated by E. C. Milner
Research supported in part by ONR Grant N00014-91-J-1150.
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Duffus, D., Goddard, T. Products of chains with monochromatic maximal chains and antichains. Order 13, 101–117 (1996). https://doi.org/10.1007/BF00389835
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DOI: https://doi.org/10.1007/BF00389835