Discrete Mathematics with Graph Theory (3rd Edition) 182

Discrete Mathematics with Graph Theory (3rd Edition) 182 -...

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Unformatted text preview: 180 Solutions to Exercises 27. (a) [BB] Focus on one of the ten people, say Hilda. Suppose first that at least four of the remaining nine people are strangers to Hilda. If these four are mutual friends we are done; otherwise two of these are strangers and, hence, together with Hilda, we have three mutual strangers and again we are done. Thus, we suppose that less than four people are strangers to Hilda; so Hilda has at least six friends in the group. By Problem 17, this group of six contains either three mutual strangers (in which case we are done) or three mutual friends who, together with Hilda, give four mutual friends. (b) Focus attention on one of the 20 people, say Bruce. Bruce has at least 10 friends among the remaining 19 people or else at least 10 people are strangers to him. If 10 are strangers, then among these there are either four mutual friends or three mutual strangers (applying part (a and if three mutual strangers, then with Bruce we get a set of four mutual strangers. So we have the if three mutual strangers, then with Bruce we get a set of four mutual strangers....
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