partitions less than cofinality

If $\\lambda<\\operatorname{cf}(\\kappa)$ then $\\kappa\\rightarrow(\\kappa)^{1}_{\\lambda}$ .

This follows easily from the definition of cofinality. For any coloring $f:\\kappa\\rightarrow\\lambda$ then define $g:\\lambda\\rightarrow\\kappa+1$ by $g(\\alpha)=|f^{-1}(\\alpha)|$ . Then $\\kappa=\\sum_{\\alpha<\\lambda}g(\\alpha)$ , and by the normal rules of cardinal arithmetic $\\operatorname{sup}_{\\alpha<\\lambda}g(\\alpha)=\\kappa$ . Since $\\lambda<\\operatorname{cf}(\\kappa)$ , there must be some $\\alpha<\\lambda$ such that $g(\\alpha)=\\kappa$ .

单词	PartitionsLessThanCofinality
释义	partitions less than cofinality If $\\lambda<\\operatorname{cf}(\\kappa)$ then $\\kappa\\rightarrow(\\kappa)^{1}_{\\lambda}$ . This follows easily from the definition of cofinality. For any coloring $f:\\kappa\\rightarrow\\lambda$ then define $g:\\lambda\\rightarrow\\kappa+1$ by $g(\\alpha)=\|f^{-1}(\\alpha)\|$ . Then $\\kappa=\\sum_{\\alpha<\\lambda}g(\\alpha)$ , and by the normal rules of cardinal arithmetic $\\operatorname{sup}_{\\alpha<\\lambda}g(\\alpha)=\\kappa$ . Since $\\lambda<\\operatorname{cf}(\\kappa)$ , there must be some $\\alpha<\\lambda$ such that $g(\\alpha)=\\kappa$ .
随便看	EuclidsProofOfTheInfinitudeOfPrimes Euclid’s algorithm Euclid’s coefficients Euclid’s lemma Euclid’s lemma proof Euclid’s proof of the infinitude of primes EulerCharacteristic Euler characteristic EulerCircuit Euler circuit EulerFermatTheorem Euler-Fermat theorem EulerFoursquareIdentity Euler four-square identity EulerGompertzConstant Euler-Gompertz constant EulerianPoset Eulerian poset EulerLagrangeDifferentialEquationadvanced Euler-Lagrange differential equation (advanced) EulerLagrangeDifferentialEquationelementary Euler-Lagrange differential equation (elementary) EulerLeonhard Euler, Leonhard EulerLine