Antichains in products of linear orders

1999/02/08 by Martin Goldstern, Saharon Shelah, Goldstern, Martin +1
Mathematics · #06A05 #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Primary 03E35 #math.GN #math.LO #msc:03E04 #msc:03E35 #msc:06A05 #secondary 03E04

paper · pdf · doi:10.48550/arxiv.math/9902054

published as Order 19 No. 3 (2002) 213--222 · 9 pages

arxiv created 1999/02/08 · arxiv updated 2009/11/30

Abstract

1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in 2,3,4, ... : There is a linear order L such that Lⁿ has no (incomparability-)antichain of cardinality lambda, while Lⁿ⁺¹ has an antichain of cardinality lambda . 2. For any nondecreasing sequence (lambda2,lambda3, ...) of infinite cardinals it is consistent that there is a linear order L such that Lⁿ has an antichain of cardinality lambda_n, but not one of cardinality lambda_n⁺ .

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