Messing with WCC

Let \([x]^\beta\) be all the \(\beta\)-element subsets of \(x\). Then an uncountable cardinal \(\alpha\) is weakly \(\beta\)-compact if and only if, for every function \(f: [\alpha]^\beta \rightarrow \{0, 1, 2\ldots, \beta'\}\) where \(\beta = \beta'+1\), there is a set \(S \subseteq \alpha\) such that \(|S| = \alpha\) and \(f\) maps every member of \([S]^\beta\) to all ordinal/cardinal \(\gamma\) where \(\gamma \in \beta\).

More intuitively, any \(\beta\)-coloring of the edges of the complete graph \(\{\beta\}K_\alpha\) contains a monochromatic subgraph isomorphic to \(\{\beta\}K_\alpha\).