Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be nice if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).
Source: 2023 Bulgaria JBMO TST Problem 4
Tags: combinatorics, set, Subset
Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be nice if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).