A Coherence of Ramsey Phenomena ∗ Ramsey number R(5, 5) = 49

The restricted coexistence as an accumulated status or an idea can be used to explain why so-called Ramsey's effects or phenomena exist. From this, some properties, for instance, for all integers q ≥ p ≥ 3, a deep inequality R(p, q) ≥ R(p − 1 ,q +2 ), are proved, based on coherence of Ramsey's phenomena and the viewpoint of accumulated quantity. Further by known data, we declare finally two Ramsey numbers R(4, 7) = R(5, 5) = 49. The theory is so unusually stunning that it spreads rapidly to all science society and influences permanently people's views toward many problems from modern science and technology, such as celebrated NP-problem from computational complexity theory or theoretic computer, completeness of axiom systems from formal logic or foundation of mathematics, and so forth. Keywords-Ramsey numbers; Restricted coexistence; Coher- ence of Ramsey phenomena; Nonuniform coexistence.