Summary: Fix an uncountable cardinal \(\lambda\). A symmetric matrix \(M=(m_{\alpha \beta})_{\alpha, \beta< \lambda}\) whose entries are countable ordinals is called strongly universal if for every positive integer \(n\), for every \(n\times n\) matrix \((b_{ij})_{i,j <n}\) and for every uncountable set \(A=\{a: a\in A\}\subseteq [\lambda]^n\) of disjoint \(n\)-tuples \(a=\{a_0, \dots, a_{n-1}\}\) there are \(a,a'\in A\) such that \(b_{ij}= m_{a_ia_j'}\) 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 undecidability 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 applications are given.