Colouring and non-productivity of \(\aleph_ 2\)-c.c (Q678760)

For cardinals \(\lambda,\mu,\theta,\sigma\) the symbol \(\text{Pr}_1(\lambda,\mu,\theta,\sigma)\) means that there exists a symmetric function \(d\colon\lambda\times\lambda\to \theta\) such that for each sequence \((u_{\alpha})_{\alpha<\mu}\) of pairwise disjoint subsets of \(\lambda\) of size less than \(\sigma\) and for every \(\gamma<\theta\) there are \(\alpha<\beta\) such that \(d[u_{\alpha}\times u_{\beta}]=\{\gamma\}\). The main result of the paper is that \(\text{Pr}_1(\aleph_2,\aleph_2,\aleph_2,\aleph_0)\) holds. Moreover, the statement \(\text{Pr}_1(\lambda^{+2},\lambda^{+2},\lambda^{+2},\lambda)\) is proved for any regular cardinal \(\lambda\).