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, │ξ│⁺<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) │f(x)∩f(y│<ℵ_α, then there is a free subset of order-type ζ for every ζ<ω_α+1.