An infinite combinatorial statement with a poset parameter

TL;DR

This paper extends classical infinite combinatorial relations using a poset parameter, providing new proofs and bounds for free set relations involving infinite cardinals, solving longstanding problems and applying order-dimension estimates.

Contribution

Introduces a new (k,l)-->P framework, offers a self-contained proof of a key relation, and derives bounds using order-dimension estimates for infinite cardinals.

Findings

Proves (l^{+n},n,l)-->n+2 for infinite cardinals and positive integers n.

Establishes (l^{+(n-1)},r,l)-->2^m with explicit bounds m.

Provides concrete examples with large aleph numbers and specific bounds.

Abstract

We introduce an extension, indexed by a partially ordered set P and cardinal numbers k,l, denoted by (k,l)-->P, of the classical relation (k,n,l)--> r in infinite combinatorics. By definition, (k,n,l)--> r holds, if every map from the n-element subsets of k to the subsets of k with less than l elements has a r-element free set. For example, Kuratowski's Free Set Theorem states that (k,n,l)-->n+1 holds iff k is larger than or equal to the n-th cardinal successor l^{+n} of the infinite cardinal k. By using the (k,l)-->P framework, we present a self-contained proof of the first author's result that (l^{+n},n,l)-->n+2, for each infinite cardinal l and each positive integer n, which solves a problem stated in the 1985 monograph of Erd\"os, Hajnal, Mate, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation (l^{+(n-1)},r,l)-->2^m, where m is the largest integer below (1/2)(1-2^{-r})^{-n/r}, for every infinite cardinal l and all positive integers n and r with r larger than 1 but smaller than n. For example, (\aleph_{210},4,\aleph_0)-->32,768. Other order-dimension estimates yield relations such as (\aleph_{109},4,\aleph_0)--> 257 (using an estimate by F\"uredi and Kahn) and (\aleph_7,4,\aleph_0)-->10 (using an exact estimate by Dushnik).