You attend a party along with N other people. Given any group of 4 party-goers you can be sure that at least one of the 4 knows the other three. Prove that at least one person knows all others at the party.

Originally posted by XanthosNZ You attend a party along with N other people. Given any group of 4 party-goers you can be sure that at least one of the 4 knows the other three. Prove that at least one person knows all others at the party.

Originally posted by XanthosNZ You attend a party along with N other people. Given any group of 4 party-goers you can be sure that at least one of the 4 knows the other three. Prove that at least one person knows all others at the party.

EDIT: I have saved this thread.

Why save a chesskid thread? Why didn't you make another one? WHY!? ðŸ˜€

assume that no guest knows all the other guests. then each guest does not know (at least) one other person. since in any group of 4 people you must know at least 1 there are a maximum of 2 strangers per guest. now according to our assumption if i pick a random guest A from the party there Exists guest B such that A,B are strangers. now we pick a third guest C if A,B,C are all strangers then they've reached their maximum of two strangers each so any fourth guest D must know them all (If there are only 4 guests then we are done) but again according to our assumption there Exists a fifth guest E who is a stranger to D. take the group A,B,D,E. A is a stranger to B who is a stranger to A and D is a stranger to E who is a stranger to D, this contradicts the assumption so at least one guest knows everyone.

in fact this shows that if A,B,C are all strangers then all the other guests know everyone at the party.