Universal matrices and strongly unbounded functions

Author(s)

Koszmider‚ P.

• Document type: Journal article
• Source: MATHEMATICAL RESEARCH LETTERS; v. 9, n. 4, p. 549-566, 2002.

Abstract

Fix an uncountable cardinal lambda. Asymmetric matrix M = (m(alphabeta))(alpha,beta<lambda) whose entries are countable ordinals is called strongly universal if for every positive integer n, for every n x n matrix (b(ij))(i,j<n) and for every uncountable set A = {a : a is an element of A} subset of or equal to [lambda](n) of disjoint n-tuples a = {a(0),...,a(n-1)}< there are a, a' is an element of A such that b(ij) = m(ai a'j) for 0 less than or equal to i, j < n. We go beyond the recent dramatic discoveries for lambda = w(1), w(2) and address the question of the possibility of the existence of a strongly universal matrix for lambda > w(2). Due to the undecidibility of some weak versions of the Ramsey property for lambda greater than or equal to w(2) the positive answer can be at most consistent, but we show that some natural methods of forcing cannot yield that answer for lambda > w(2). We use our method of "forcing with side conditions in semimorasses" to construct generically lambda by lambda strongly universal matrices for any cardinal lambda. The results are proved in more generality, related concepts are investigated, some questions are stated and some application are given. (AU)

• FAPESP's process: 97/10491-7 - Piotr Koszmider | Ohio State University - Estados Unidos
• Grantee: Lúcia Renato Junqueira
• Support Opportunities: Research Grants - Visiting Researcher Grant - International