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Nombres de Ramsey

Envoyé par babsgueye 
Nombres de Ramsey
il y a huit mois
avatar
Salut

Je pense qu'on peut poster ce sujet ici et non en arithmétique, sinon...
J'ai un petit problème avec les nombres de Ramsey et les notes que je lis sur Wikipedia. Je cherche à trouver $R(5, 5)$ qui est supposé (c'est démontrer) être entre $43$ et $48$ compris, mais j'ai fait une excursion ailleurs.
D'après le tableau des nombres de Ramsey de Wikipedia $R(6, 5)$ est compris entre $58$ et $87$. Ce qui me fait penser que en général $R(r, s)\neq R(s, r)$.
Mais avec le graphe suivant à $48$ sommets, qui donne que les ''arètes bleues'', que j'ai obtenu, je ne comprends plus vraiment le tableau de Wikipédia, particulièrement la valeur de $R(6, 5)$.

Le graphe :

V := Graph(undirected, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 9}, {1, 18}, {1, 19}, {1, 20}, {1, 22}, {1, 23}, {1, 24}, {1, 25}, {1, 30}, {1, 31}, {1, 32}, {1, 34}, {1, 35}, {1, 36}, {1, 37}, {1, 42}, {1, 43}, {1, 44}, {1, 46}, {1, 47}, {1, 48}, {2, 3}, {2, 4}, {2, 6}, {2, 10}, {2, 17}, {2, 19}, {2, 20}, {2, 21}, {2, 23}, {2, 24}, {2, 26}, {2, 29}, {2, 31}, {2, 32}, {2, 33}, {2, 35}, {2, 36}, {2, 38}, {2, 41}, {2, 43}, {2, 44}, {2, 45}, {2, 47}, {2, 48}, {3, 4}, {3, 7}, {3, 11}, {3, 17}, {3, 18}, {3, 20}, {3, 21}, {3, 22}, {3, 24}, {3, 27}, {3, 29}, {3, 30}, {3, 32}, {3, 33}, {3, 34}, {3, 36}, {3, 39}, {3, 41}, {3, 42}, {3, 44}, {3, 45}, {3, 46}, {3, 48}, {4, 8}, {4, 12}, {4, 17}, {4, 18}, {4, 19}, {4, 21}, {4, 22}, {4, 23}, {4, 28}, {4, 29}, {4, 30}, {4, 31}, {4, 33}, {4, 34}, {4, 35}, {4, 40}, {4, 41}, {4, 42}, {4, 43}, {4, 45}, {4, 46}, {4, 47}, {5, 6}, {5, 7}, {5, 8}, {5, 9}, {5, 14}, {5, 15}, {5, 16}, {5, 22}, {5, 23}, {5, 24}, {5, 26}, {5, 27}, {5, 28}, {5, 29}, {5, 34}, {5, 35}, {5, 36}, {5, 38}, {5, 39}, {5, 40}, {5, 41}, {5, 46}, {5, 47}, {5, 48}, {6, 7}, {6, 8}, {6, 10}, {6, 13}, {6, 15}, {6, 16}, {6, 21}, {6, 23}, {6, 24}, {6, 25}, {6, 27}, {6, 28}, {6, 30}, {6, 33}, {6, 35}, {6, 36}, {6, 37}, {6, 39}, {6, 40}, {6, 42}, {6, 45}, {6, 47}, {6, 48}, {7, 8}, {7, 11}, {7, 13}, {7, 14}, {7, 16}, {7, 21}, {7, 22}, {7, 24}, {7, 25}, {7, 26}, {7, 28}, {7, 31}, {7, 33}, {7, 34}, {7, 36}, {7, 37}, {7, 38}, {7, 40}, {7, 43}, {7, 45}, {7, 46}, {7, 48}, {8, 12}, {8, 13}, {8, 14}, {8, 15}, {8, 21}, {8, 22}, {8, 23}, {8, 25}, {8, 26}, {8, 27}, {8, 32}, {8, 33}, {8, 34}, {8, 35}, {8, 37}, {8, 38}, {8, 39}, {8, 44}, {8, 45}, {8, 46}, {8, 47}, {9, 10}, {9, 11}, {9, 12}, {9, 14}, {9, 15}, {9, 16}, {9, 18}, {9, 19}, {9, 20}, {9, 26}, {9, 27}, {9, 28}, {9, 30}, {9, 31}, {9, 32}, {9, 33}, {9, 38}, {9, 39}, {9, 40}, {9, 42}, {9, 43}, {9, 44}, {9, 45}, {10, 11}, {10, 12}, {10, 13}, {10, 15}, {10, 16}, {10, 17}, {10, 19}, {10, 20}, {10, 25}, {10, 27}, {10, 28}, {10, 29}, {10, 31}, {10, 32}, {10, 34}, {10, 37}, {10, 39}, {10, 40}, {10, 41}, {10, 43}, {10, 44}, {10, 46}, {11, 12}, {11, 13}, {11, 14}, {11, 16}, {11, 17}, {11, 18}, {11, 20}, {11, 25}, {11, 26}, {11, 28}, {11, 29}, {11, 30}, {11, 32}, {11, 35}, {11, 37}, {11, 38}, {11, 40}, {11, 41}, {11, 42}, {11, 44}, {11, 47}, {12, 13}, {12, 14}, {12, 15}, {12, 17}, {12, 18}, {12, 19}, {12, 25}, {12, 26}, {12, 27}, {12, 29}, {12, 30}, {12, 31}, {12, 36}, {12, 37}, {12, 38}, {12, 39}, {12, 41}, {12, 42}, {12, 43}, {12, 48}, {13, 14}, {13, 15}, {13, 16}, {13, 17}, {13, 21}, {13, 25}, {13, 30}, {13, 31}, {13, 32}, {13, 34}, {13, 35}, {13, 36}, {13, 37}, {13, 42}, {13, 43}, {13, 44}, {13, 46}, {13, 47}, {13, 48}, {14, 15}, {14, 16}, {14, 18}, {14, 22}, {14, 26}, {14, 29}, {14, 31}, {14, 32}, {14, 33}, {14, 35}, {14, 36}, {14, 38}, {14, 41}, {14, 43}, {14, 44}, {14, 45}, {14, 47}, {14, 48}, {15, 16}, {15, 19}, {15, 23}, {15, 27}, {15, 29}, {15, 30}, {15, 32}, {15, 33}, {15, 34}, {15, 36}, {15, 39}, {15, 41}, {15, 42}, {15, 44}, {15, 45}, {15, 46}, {15, 48}, {16, 20}, {16, 24}, {16, 28}, {16, 29}, {16, 30}, {16, 31}, {16, 33}, {16, 34}, {16, 35}, {16, 40}, {16, 41}, {16, 42}, {16, 43}, {16, 45}, {16, 46}, {16, 47}, {17, 18}, {17, 19}, {17, 20}, {17, 21}, {17, 26}, {17, 27}, {17, 28}, {17, 29}, {17, 34}, {17, 35}, {17, 36}, {17, 38}, {17, 39}, {17, 40}, {17, 41}, {17, 46}, {17, 47}, {17, 48}, {18, 19}, {18, 20}, {18, 22}, {18, 25}, {18, 27}, {18, 28}, {18, 30}, {18, 33}, {18, 35}, {18, 36}, {18, 37}, {18, 39}, {18, 40}, {18, 42}, {18, 45}, {18, 47}, {18, 48}, {19, 20}, {19, 23}, {19, 25}, {19, 26}, {19, 28}, {19, 31}, {19, 33}, {19, 34}, {19, 36}, {19, 37}, {19, 38}, {19, 40}, {19, 43}, {19, 45}, {19, 46}, {19, 48}, {20, 24}, {20, 25}, {20, 26}, {20, 27}, {20, 32}, {20, 33}, {20, 34}, {20, 35}, {20, 37}, {20, 38}, {20, 39}, {20, 44}, {20, 45}, {20, 46}, {20, 47}, {21, 22}, {21, 23}, {21, 24}, {21, 26}, {21, 27}, {21, 28}, {21, 30}, {21, 31}, {21, 32}, {21, 33}, {21, 38}, {21, 39}, {21, 40}, {21, 42}, {21, 43}, {21, 44}, {21, 45}, {22, 23}, {22, 24}, {22, 25}, {22, 27}, {22, 28}, {22, 29}, {22, 31}, {22, 32}, {22, 34}, {22, 37}, {22, 39}, {22, 40}, {22, 41}, {22, 43}, {22, 44}, {22, 46}, {23, 24}, {23, 25}, {23, 26}, {23, 28}, {23, 29}, {23, 30}, {23, 32}, {23, 35}, {23, 37}, {23, 38}, {23, 40}, {23, 41}, {23, 42}, {23, 44}, {23, 47}, {24, 25}, {24, 26}, {24, 27}, {24, 29}, {24, 30}, {24, 31}, {24, 36}, {24, 37}, {24, 38}, {24, 39}, {24, 41}, {24, 42}, {24, 43}, {24, 48}, {25, 26}, {25, 27}, {25, 28}, {25, 29}, {25, 33}, {25, 42}, {25, 43}, {25, 44}, {25, 46}, {25, 47}, {25, 48}, {26, 27}, {26, 28}, {26, 30}, {26, 34}, {26, 41}, {26, 43}, {26, 44}, {26, 45}, {26, 47}, {26, 48}, {27, 28}, {27, 31}, {27, 35}, {27, 41}, {27, 42}, {27, 44}, {27, 45}, {27, 46}, {27, 48}, {28, 32}, {28, 36}, {28, 41}, {28, 42}, {28, 43}, {28, 45}, {28, 46}, {28, 47}, {29, 30}, {29, 31}, {29, 32}, {29, 33}, {29, 38}, {29, 39}, {29, 40}, {29, 46}, {29, 47}, {29, 48}, {30, 31}, {30, 32}, {30, 34}, {30, 37}, {30, 39}, {30, 40}, {30, 45}, {30, 47}, {30, 48}, {31, 32}, {31, 35}, {31, 37}, {31, 38}, {31, 40}, {31, 45}, {31, 46}, {31, 48}, {32, 36}, {32, 37}, {32, 38}, {32, 39}, {32, 45}, {32, 46}, {32, 47}, {33, 34}, {33, 35}, {33, 36}, {33, 38}, {33, 39}, {33, 40}, {33, 42}, {33, 43}, {33, 44}, {34, 35}, {34, 36}, {34, 37}, {34, 39}, {34, 40}, {34, 41}, {34, 43}, {34, 44}, {35, 36}, {35, 37}, {35, 38}, {35, 40}, {35, 41}, {35, 42}, {35, 44}, {36, 37}, {36, 38}, {36, 39}, {36, 41}, {36, 42}, {36, 43}, {37, 38}, {37, 39}, {37, 40}, {37, 41}, {37, 45}, {38, 39}, {38, 40}, {38, 42}, {38, 46}, {39, 40}, {39, 43}, {39, 47}, {40, 44}, {40, 48}, {41, 42}, {41, 43}, {41, 44}, {41, 45}, {42, 43}, {42, 44}, {42, 46}, {43, 44}, {43, 47}, {44, 48}, {45, 46}, {45, 47}, {45, 48}, {46, 47}, {46, 48}, {47, 48}}, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48])

Ce graphe ne contient pas de $K_6$ et son complémentaire ne contient pas de $K_5$ !

Quelqu'un peut-il lever mon incompréhension ? Merci.

''Dans un point, il n'y a pas de matière, donc il y a de l'esprit, et que de l'esprit.''
Re: Nombres de Ramsey
il y a huit mois
avatar
Ah, je vois. En fait j'avais écris ma réponse. C'est parce qu'en général $R(s, r)\neq R(r, s)$.....!

Excuse.
Re: Nombres de Ramsey
il y a huit mois
avatar
J'ai encore besoin de quelques détails, parce que d'après Wikipedia $R(s, r) = R(r, s)$, et donc à mon avis, le théorème n'est pas très clair dans le cas où $r\neq s$.
Re: Nombres de Ramsey
il y a huit mois
avatar
C'est bon, j'ai vu la nuance. Excusez-moi d'avoir trop vite poste
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