Partition of Infinity

1. Partition of Infinity Nguyen Duy Long, Liu Zhe Mentor: Mr Ku Ching Ming Abstract : To investigate the properties of Infinity through Partition relation. α κ→﴾λ﴿β For set L with |L|=κ and every partition of [L] α into β classes, there is a homogeneous set M with |M| > λ / 2 6 → (3)2 ↕ R(3,3)=6 א0 α+1 2 א1 → (א1)2 2 = א / (Continuum Hypothesis) Set Theory m |R| > א0 א0 → (א0)n (Cantor’s Theorem) • If κ→﴾λ﴿β is true for some κ, β, λ, α, then • 1. κ+→﴾λ﴿β • 2. κ→﴾λ-﴿β • 3. κ→﴾λ﴿β • 4. κ→﴾λ﴿β ` א1 → (א1,א0)2 ? α α α - - α 2