Antichains, the stick principle, and a matching number

摘要

We investigate basic properties of three cardinal invariants involving w(1): stick, antichain number, and matching number. The antichain number is the least cardinal kappa for which there does not exist a subcollection of size kappa of uncountable subsets of omega(1) with pairwise finite intersections and the matching number is the least cardinal a for which there exists a subcollection X of size kappa of order-type omega subsets of omega(1) so that every uncountable subset of omega(1) has infinite intersection with a member of X. We demonstrate how these numbers are affected by Cohen forcing and also prove some results about the effect of Hechler forcing. We also introduce a forcing notion to increase the matching number, and study its basic properties. (C) 2019 Elsevier B.V. All rights reserved.