Show that in any finite gathering of people, there are at least two people who know...

Show that in any finite gathering of people, there are at least two people who know the same number of people at the gathering (assume that “knowing” is a mutual relationship). Hint available.

Expert Solution

Lets recall the Pigeon hole principle, it says if we have n+1 pigeon and n holes and we have to put the all pigeon in the holes then there will be at least one hole in which there are two pigeon.

Now lets come to the problem , let we have n no of people in the gathering then there is two case

case -1 there is one person who knows everyone that means each one knows at least one person(because "knowing" is a mutual relation ) ,so the set that represent the possible no of people one can know is {1,2,......,n-1},but we have n people so by Pigeon hole principle there will be at least two person who know the same no of people.

case-2 there is a person who does not know anyone that means each people can know at most n-2 people ,so out set of possibility is given as {0,1,2,.......,n-2} ,but we have n people so again using Pigeon hole principle there will be at least two people who know same no of people.

Colby Messinger answered 2 years ago

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