Problem

Source: 2018 China Southeast MO Grade 11 P6

Tags: combinatorics



Assume integer $m \geq 2.$ There are $3m$ people in a meeting, any two of them either shake hands with each other once or not.We call the meeting "$n$-interesting", only if there exists $n(n\leq 3m-1)$ people of them, the time everyone of whom shakes hands with other $3m-1$ people is exactly $1,2,\cdots,n,$ respectively. If in any "$n$-interesting" meeting, there exists $3$ people of them who shake hands with each other, find the minimum value of $n.$