Measurable cardinals are weakly compact

Let U be a normal measure on kappa, supplied by F1, and fix a coloring $c:[\kappa {]}^2\to 2$. For each alpha<kappa the tail above alpha is measure one by F2. Its two color fibres are disjoint and cover that tail; exactly one belongs to U by the ultrafilter laws. Let i_alpha be this uniquely determined color and A_alpha its fibre. Exactly one of the sets $H_i=\{\alpha :i_{\alpha }=i\}$, i<2, belongs to U. Fix its color i. These selections are unique finite decisions, with no extra choice beyond the ambient F4.

By F1 the diagonal intersection $D=\{\beta <\kappa :(\mathrm{\forall }\alpha <\beta )\beta \in A_{\alpha }\}$ belongs to U. Thus H=H_i intersect D belongs to U and has size kappa by F2. If alpha<beta both belong to H, then alpha lies in H_i and beta lies in D, so beta belongs to A_alpha and c(alpha,beta)=i_alpha=i. Hence H is homogeneous. F2 makes kappa inaccessible, and F3 turns this two-color partition property into weak compactness.