Universal matrices and strongly unbounded functions

Fix an uncountable cardinal $\lambda$. A symmetric matrix $M=(m_{\alpha \beta})_{\alpha,\beta<\lambda}$ whose entries are countable ordinals is called {\it strongly universal} if for every positive integer $n$, for every $n\times n$ matrix $(b_{i j})_{i,j<n}$ and for every uncountable set $A=\{a: a\in A\}\subseteq[\lambda]^n$ of disjoint $n$-tuples $a=\{a_0,...,a_{n-1}\}_<$ there are $a, a'\in A$ such that $b_{ij}=m_{a_i a_j'}$ for $0\leq i, j<n$. We go beyond the recent dramatic discoveries for $\lambda=\omega_1, \omega_2$ and address the question of the possibility of the existence of a strongly universal matrix for $\lambda >\omega_2$. Due to the undecidibility of some weak versions of the Ramsey property for $\lambda\geq\omega_2$ the positive answer can be at most consistent, but we show that some natural methods of forcing cannot yield that answer for $\lambda >\omega_2$. 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.

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Published 1 January 2002