An infinite combinatorial statement with a poset parameter

Abstract

We introduce an extension, indexed by a partially ordered set P and cardinal numbers κ,λ, denoted by (κ,<λ)⇝P, of the classical relation (κ,n,λ)→ρ in infinite combinatorics. By definition, (κ,n,λ)→ρ holds if every map F: [κ]n→[κ]<λ has a ρ-element free set. For example, Kuratowski’s Free Set Theorem states that (κ,n,λ)→n+1 holds iff κλ +n, where λ +n denotes the n-th cardinal successor of an infinite cardinal λ. By using the (κ,<λ)⇝P framework, we present a self-contained proof of the first author’s result that (λ +n,n,λ)→n+2, for each infinite cardinal λ and each positive integer n, which solves a problem stated in the 1985 monograph of Erdős, Hajnal, Máté, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation \((\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1} {2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor } \) , for every infinite cardinal λ and all positive integers n and r with 2≤r<n. For example, (ℵ210,4,ℵ0)→32,768. Other order-dimension estimates yield relations such as (ℵ109,4,ℵ0) → 257 (using an estimate by Füredi and Kahn) and (ℵ7,4,ℵ0)→10 (using an exact estimate by Dushnik).

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Correspondence to Pierre Gillibert.

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This work was partially supported by the institutional grant MSM 0021620839.

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Gillibert, P., Wehrung, F. An infinite combinatorial statement with a poset parameter. Combinatorica 31, 183 (2011). https://doi.org/10.1007/s00493-011-2602-y

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Mathematics Subject Classification (2000)

  • 03E05
  • 06A07
  • 05A05
  • 05D10
  • 06A05