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A multicolour Ramsey number is a Ramsey number using 3 or more colours. There are (up to symmetries) only two non-trivial multicolour Ramsey numbers for which the exact value is known, namely R(3, 3, 3) = 17 and R(3, 3, 4) = 30. [1] Suppose that we have an edge colouring of a complete graph using 3 colours, red, green and blue.
There are two ways to track the effect of changing R to f(R), i.e., to obtain the theory field equations. The first is to use metric formalism and the second is to use the Palatini formalism. [3] While the two formalisms lead to the same field equations for General Relativity, i.e., when f(R) = R, the field equations may differ when f(R) ≠ R.
The Harrison topology is a topology on the set of orderings X F of a formally real field F. Each order can be regarded as a multiplicative group homomorphism from F ∗ onto ±1. Giving ±1 the discrete topology and ±1 F the product topology induces the subspace topology on X F.
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An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
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On the other hand, the completions with respect to the other non-trivial absolute values give the fields of p-adic numbers, where is a prime integer number (see below); since the -adic absolute values satisfy the ultrametric property, then the -adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields).
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