Notes on combinatorial set theory

We shall prove some unconnected theorems: (1) (G.C.H.) \omega _{\alpha + 1} \to \left( {\omega _\alpha + \xi } \right)_2^2 when א_α is regular, {box drawings light vertical}ξ{box drawings light vertical}⁺<ωα. (2) There is a Jonsson algebra in א_α+n, and \aleph _{a + n} \not \to \left[ {\aleph _{a + n} } \right]_{\aleph _{a + n} }^{n + 1} if 2^{\aleph _{ - - } } = \aleph _{a + n} \cdot (3) If λ>א₀ is a strong limit cardinal, then among the graphs with ≦λ vertices each of valence <λ there is a universal one. (4)(G.C.H.) If f is a set mapping on \omega _{a + 1} (א_α regular) {box drawings light vertical}f(x)∩f(y{box drawings light vertical}<א_α, then there is a free subset of order-type ζ for every ζ<ω_α+1.