caa a22 4500 445358726 CHVBK 20180317142908.0 cr unu---uuuuu 170323e20110301xx s 000 0 eng 10.1007/s00493-011-2602-y doi (NATIONALLICENCE)springer-10.1007/s00493-011-2602-y An infinite combinatorial statement with a poset parameter [Elektronische Daten] [Pierre Gillibert, Friedrich Wehrung] 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 κ ≥ λ +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). János Bolyai Mathematical Society and Springer Verlag, 2011 Gillibert Pierre LMNO, CNRS UMR 6139 Département de Mathématiques, BP 5186, Université de Caen, Campus 2, 14032, Caen cedex, France aut Wehrung Friedrich LMNO, CNRS UMR 6139 Département de Mathématiques, BP 5186, Université de Caen, Campus 2, 14032, Caen cedex, France aut Combinatorica Springer-Verlag 31/2(2011-03-01), 183-200 0209-9683 31:2<183 2011 31 493 https://doi.org/10.1007/s00493-011-2602-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00493-011-2602-y text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gillibert Pierre LMNO, CNRS UMR 6139 Département de Mathématiques, BP 5186, Université de Caen, Campus 2, 14032, Caen cedex, France aut NATIONALLICENCE 700 1- Wehrung Friedrich LMNO, CNRS UMR 6139 Département de Mathématiques, BP 5186, Université de Caen, Campus 2, 14032, Caen cedex, France aut NATIONALLICENCE 773 0- Combinatorica Springer-Verlag 31/2(2011-03-01), 183-200 0209-9683 31:2<183 2011 31 493 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer