Erdős problem 70

Trivial boundary case: $\mathfrak{c} \to (\omega, 3)^3_2$.

This is trivially true because in a 3-uniform hypergraph, a \"blue clique of size 3\" consists of a single 3-element subset ($\binom{3}{3} = 1$), so the blue alternative merely asks for one blue triple to exist. The proof splits into two cases: - If any blue triple exists, it is itself a blue-monochromatic set of cardinality 3. - If no blue triple exists, all triples are red, and since $\omega \le \mathfrak{c}$, any subset of order type $\omega$ is red-monochromatic.

The problem becomes non-trivial only for $n \ge 4$; see omega_times_two_four for the simplest genuinely open case.