Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 130 / 3 / b

Write ${\forall }_{\mathcal{U}}x\phi (x)$ to mean $\{x:\phi (x)\}\in \mathcal{U}$. For each $x$, exactly one of the red-neighbour set $R_x$ and the blue-neighbour set $B_x$ belongs to $\mathcal{U}$. Applying the ultrafilter dichotomy once more toshows that exactly one ofholds. They cannot both hold because the two outer sets are complementary; the diagonal causes no problem because a nonprincipal ultrafilter contains no singleton.

Assume the red statement and put $X=\{x:R_x\in \mathcal{U}\}\in \mathcal{U}$. Choose $x_1\in X$. Recursively chooseoutside the finitely many previously chosen points. Every set in this finite intersection belongs to $\mathcal{U}$, so a choice is always possible. Then $M=\{x_1,x_2,\dots \}$ is infinite and all of its edges are red.

Solved by gpt-5.6-sol high.